Approximation Methods in QM

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Approximation Methods in QM"

Transcription

1 Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate) Degenerate perturbation theory (PT) Time dependent PT and Fermi s golden rule Variational method and helium atom Basic Questions A. What causes fine strucutre in hydrogen spectra? B. What is Fermi s golden rule? C. What is the variational principle? 51

2 5 CHAPTER 3. APPROXIMATION METHODS IN QM 3.1 Time-independent PT (nondegenerate) Introduction Two quantum problems can be solved exactly, they are harmonic oscillator and hydrogen (or hydrogen-lie) atom. Most other quantum problems can not be solved exactly. One has to develop approximate method to solve such problem. Consider a Hamiltonian Ĥ, its Schrödinger eq. ĤΨ p E p Ψ p, is difficult to solve. If Ĥ can be written as Ĥ Ĥ0 + ˆV p quantum no. where ˆV is time-independent and small, and the Schrödinger eq. of Ĥ 0 has been solved Ĥ 0 Φ e Φ Ĥ0 Φ e Φ where is the quantum number labelling the nondegenerate states. We wish to use the information of Ĥ 0 to find the eigenstates and eigenvalues of Ĥ. Ĥ 0 is usually referred to as unperturbed Hamiltonian, ˆV as perturbation potential (operator). In principle (the completeness principle), any state Ψ p can be written as a linear combination of Φ as Ψ p C p Φ where {C p } are constants to be determined by eq. of Ĥ, together with eigenvalue E p. Since the dimensionality (total number of the possible ) is in general infinite, the exact solution is not easy. However, since the perturbation ˆV is assumed small, it is not difficult to get approximate solution for Ψ p and E p Perturbation expansion Obviously, if ˆV is zero, p, Ψ Φ, E e. Hence, for non-zero ˆV, this solution {p, Ψ Φ, E e } is referred to as zero-order perturbation results. We need to go beyond this, and find nontrivial correction to this zero-order results. In order to do it in a systematic way, we introduce a parameter λ in the Hamiltonian Ĥ (λ) Ĥ0 + λ ˆV, 0 λ 1 so that as λ 0, solutions for Ĥ (λ) is reduced to that of Ĥ 0 and what we want is the results in the limit λ 1. In perturbation theory, we assume p

3 3.1. TIME-INDEPENDENT PT (NONDEGENERATE) 53 i.e., the quantum number for the states of Ĥ is same as that of Ĥ 0. Therefore, we can expand each of the eigenvalues and eigenstates of Ĥ (λ) in terms of powers of λ as E E (0) + λe (1) + λ E () + λ 3 E (3) + Ψ Ψ (0) + λψ (1) + λ Ψ () + λ 3 Ψ (3) + where E (0) e is the zero-order approximation to the -th eigenvalue of Ĥ, and E (1), E(), E(3), are the successive correction to this. Similarly, Ψ(0) Φ is the zero-order appro. to -th eigenstate of Ĥ and Ψ (1), Ψ(), Ψ(3), are the successive correction to this. In order to find E (1), E(), E(3), and Ψ(1), Ψ(), Ψ(3),, we substitute the expansions into the original Schrödinger eq. ĤΨ E Ψ or (Ĥ0 + λ ˆV ) ( Ψ (0) + λψ (1) + λ Ψ () + ) ( E (0) + λe (1) + λ E () + ) ( Ψ (0) + λψ (1) + λ Ψ () + ) or, each side is arranged according to the powers of λ Ĥ 0 Ψ (0) + λ ( ˆV Ψ (0) + E (0) Ψ(0) + λ ( E (1) Ψ(0) ) (1) Ĥ0Ψ ( + λ (1) ˆV Ψ + E (0) Ψ(1) + ) ( + λ E () Ψ(0) ) () Ĥ0Ψ + + E (1) Ψ(1) + E (0) ) Ψ() + Equating the coefficients of λ on both sides of the eq. (consider λ as an arbitrary number, this must be true), we have as zero-order result, and Ĥ 0 Ψ (0) E (0) Ψ(0) Ψ (0) Φ, E (0) e ˆV Ψ (0) (1) + Ĥ0Ψ E (1) as first-order eq. for E (1) and Ψ (1) and ˆV Ψ (1) () + Ĥ0Ψ E () as second-order eq. for E () and Ψ () etc. Ψ(0) Ψ(0) + E (1) + E (0) Ψ(1) Ψ(1) + E (0) Ψ()

4 54 CHAPTER 3. APPROXIMATION METHODS IN QM Solution for perturbation eq.: 1st-order Now consider the solution to the above sequence of the perturbation eqs. The 1storder eq. is ˆV Ψ (0) (1) + Ĥ0Ψ E (1) Ψ(0) + E (0) Ψ(1) or, using the zero-order results ( ) (1) ˆV E Φ ( ) e Ĥ0 (1) Ψ in terms of a linear com- By the completeness principle, we can always express Ψ (1) bination of Φ as Ψ (1) C (1) Φ where C (1) are coefficients to be determined and the equation becomes, using Ĥ0Φ e Φ ( ) (1) ˆV E Φ ( ˆV E (1) ) Φ (e e ) C (1) Φ (e e ) C (1) Φ this is the eq. for E (1) and C. The standard technique to solve this ind eq. is to tae inner product with state Φ on both sides of the eq., i.e., multiplying Φ and integrate using the orthonormal relation we have ( ) d 3 rφ (1) ˆV E Φ (e e ) C (1) d 3 rφ Φ δ d 3 rφ Φ V E (1) δ (e e ) C (1) δ (e e ) C (1) V d 3 rφ ˆV Φ Choosing we have E (1) V d 3 rφ ˆV Φ

5 3.1. TIME-INDEPENDENT PT (NONDEGENERATE) 55 Choosing we have C (1) V e e Note that C is still undetermined and can be chosen as zero. The 1st-order results are, letting λ 1, E e + E (1) V. d 3 rφ ˆV Φ for the energy and V Ψ Φ + e e Φ for the state. Note: if we choose C 0, the coefficient of Φ in the above formula will not be unity and Ψ is then not properly normalized in the limit V 0. Note that the determine of the 1st-order energy E (1) only depends on the zeroorder state function Ψ (0) Φ, and the 1st-order statefunction Ψ (1) is not necessary. However, in order to determine the nd -order energy correction E (), the 1st-order statefunction correction Ψ (1) is required. This is generally true: nth-order energy correction requires the (n 1)th-order statefunction correction Solution for perturbation eq.: st-order The nd-order equation is ˆV Ψ (1) () + Ĥ0Ψ E () Ψ(0) + E (1) Ψ(1) + E (0) Ψ() or E () Ψ(0) ( ) (1) (1) ˆV E Ψ + ( Ĥ 0 E (0) ) () Ψ where E (0) Again, Ψ (), E(1), Ψ(0) and Ψ (1) are nown. We only need to determine E () and Ψ (). can be written as a linear combination of Φ Ψ () C () Φ substituting this into the eq., using the zero- and 1st-order results, we have E () Φ C (1) ( ) (1) ˆV E Φ + C () ) (Ĥ0 e Φ C (1) ( (1) ˆV E ) Φ + C () (e e ) Φ

6 56 CHAPTER 3. APPROXIMATION METHODS IN QM as before, taing inner product with state Φ on both sides of the eq. and using the orthonormality of Φ we have E () δ C (1) ( ˆV E(1) δ ) + C () (e e ). Choosing, we obtain E () E () C (1) ( V E (1) δ ) C (1) V, C(1) 0 V V e e V e e where we have assume ˆV is a Hermitian operator. The coefficient C () mined by choosing, as before. Again, to determine E () C (1) and C() is not necessary. The nd-order result for the energy is then given by E E (0) + E (1) + E () e + V + Ψ Ψ (0) + Ψ (1) Φ + V e e Φ can be deter- we only need to now V e e with V ˆV d 3 r Φ ˆV Φ Example Consider an anharmonic oscillator with Hamiltonian Ĥ Ĥ0 + ˆV Ĥ 0 h m x + 1 mωx, ˆV αx 3 where α is a small number (α hω/x 3 0 hω ( ) mω 3/). h The eigenequation of Ĥ 0 has been solved, ( Ĥ 0 n hω n + 1 ) n, n 0, 1,,, ( Ĥ 0 hω â â + 1 )

7 3.1. TIME-INDEPENDENT PT (NONDEGENERATE) 57 and we now the effect of â and â on the normalized state n â n n + 1 n + 1, â n n n 1 for n > 0. we hence need to express ˆV in terms of â and â. Using the definition â mω h ˆx + iˆp, m hω â mω h ˆx iˆp m hω we have hence ˆx h (â + â ), ˆp 1 i mω h (â â ) ˆV ( ) 3/ h (â α + â 3 ( ) 3/ h (â α + â â + ââ + â â + â ) ( ) 3/ h (â α + â â + â â â ). Up to nd-order approximation, we have E n e n + V nn + n n V nn V n n e n e n where ( e n hω n + 1 ) and V nn n ˆV n ( ) 3/ h α n ( â + â ) ( â + â â â ) n ( ) 3/ h α n ( â + â ) ( â + n â ) n 0 i.e., the first-order energy correction is zero.

8 58 CHAPTER 3. APPROXIMATION METHODS IN QM To determine the nd-order energy correction, we need to evaluate V n n. Consider (â + â ) ( â + â â â ) n ( â + â ) ( â + n â ) n ( â + â ) ( ) n (n 1) n + (n + 1) n + (n + 1) (n + ) n + n (n 1) (n ) n 3 + (n + 1) n n 1 + (n + 1) (n + ) n n (n 1) n 1 + (n + 1) n + 1 n (n + 1) (n + ) (n + 3) n + 3 [ n (n 1) (n ) n 3 + (n + 1) ] n + n (n 1) n 1 + [ + (n + 1) (n + ) + (n + 1) ] n + 1 n (n + 1) (n + ) (n + 3) n + 3 n (n 1) (n ) n 3 + 3n 3/ n (n + 1) 3/ n (n + 1) (n + ) (n + 3) n + 3 therefore, there are only four nonzero contribution to V n n therefore n n V nn V n n e n e n V n 3,n ( ) 3/ n 3 ˆV h n α n (n 1) (n ) V n 1,n ( ) 3/ n 1 ˆV h n α 3n 3/ V n+1,n ( ) 3/ n + 1 ˆV h n α 3 (n + 1) 3/ V n+3,n ( ) 3/ n + 3 ˆV h n α (n + 1) (n + ) (n + 3) V n 3,n e n e n hω α + 1 hω α 1 hω α V n 1,n + V n+1,n + V n+3,n e n e n 1 e n e n+1 e n e n+3 ( ) 3 h n (n 1) (n ) ( ) 3 h 9n 3 ( ) 3 h 9 (n + 1) 3

9 3.. DEGENERATE PT 59 1 ( h 3 hω α 1 3 hω α ( h ) 3 ) 3 (n + 1) (n + ) (n + 3) [ n (n 1) (n ) + 7n 3 7 (n + 1) 3 (n + 1) (n + ) (n + 3) ] 1 ( ) 3 h ( hω α 30n + 30n + 11 ) Hence, up to nd-order, the energy value is ( E n hω n + 1 ) 1 ( ) 3 h ( hω α 30n + 30n + 11 ) and the energy interval between adjacent energy levels See figure E n E n 1 hω 60 ( ) 3 h hω α n 3. Degenerate PT In the above perturbation theory, the unperturbed states Φ are nondegenerate, e e for all. Now we extend the theory to the degenerate case. Our Hamiltonian is still given by Ĥ Ĥ0 + ˆV where the perturbed term ˆV is assumed small and the unperturbed Hamiltonian Ĥ0 has been solved Ĥ 0 Φ nd e n Φ nd with quantum number d 1,,, M, labelling the degeneracy. In the case of Hydrogen, we now for each n, we have degenerate qn d (l, m), d l,m and M. For convenience we only need one letter for the notation. We expect that the

10 60 CHAPTER 3. APPROXIMATION METHODS IN QM introduction of perturbed term will lift the degeneracy in l, so the eigen equation for Ĥ is ĤΨ nd E nd Ψ nd. But if we still carry out the similar analysis as before, we expect to encounter a factor such as Ĥ Ĥ (λ) Ĥ0 + λ ˆV, and the expansion series E nd E (0) nd + λe(1) nd + Ψ nd Ψ (0) nd + λψ(1) nd + substituting these into the original Schrödinger eq. and equating the coefficients of λ on both sides of the eq. (consider λ as an arbitrary number, this must be true), we have Ĥ 0 Ψ (0) nd E(0) nd Ψ(0) n,d Ψ (0) nd Φ nd, E (0) nd e n as zero-order result, and ˆV Ψ (0) (1) nd + Ĥ0Ψ nd E(1) nd Ψ(0) nd + E(0) nd Ψ(1) nd as first-order eq. etc. All those formulae remain the same, only replacing single qn by double ones (n, d). However, there are divergence problem when we go beyond zeroth-order. Consider the 1st-order correction to the wavefunction Ψ (1) nd n d c (1) nd,n d Φ n d c (1) nd,nd Φ nd + other contributions from states with n n d d with the diverging coefficient c (1) nd,nd V nd,nd E (0) nd E(0) nd Solution: Recall the basic assumption in PT V nd,nd e n e n. quantum number after perturbation quantum number before perturbation. We need to modify this by introducing a new quantum number as before perturbation : (n, d) after perturbation : (n, ). This is equivalent to introduce a new basic set {Ω n, 1,,, M}as M Ω n b d Φ nd, d1 1,,, M

11 3.. DEGENERATE PT 61 with coefficients b d to be determined by the condition to avoid the divergence mentioned above. Now our new PT expansion series are E n E (0) n + λe(1) n + Ψ n Ψ (0) n + λψ(1) n + and the zeroth-order equation does not change much and the 1st-order eq. Ĥ 0 Ψ (0) n E(0) n Ψ(0) n E(0) n e n, Ψ (0) n Ω n where we expand Ψ (1) n in terms of Ω n ( ) (1) (0) ˆV E n Ψ n ( E (0) n ) Ĥ0 (1) Ψ n Ψ (1) n c (1) Ω n, 1,,, M. M b d Φ nd d1 As before, we substitute this into the 1st-order eq. and tae inner product with state Ω n on both sides, we have ( ) d 3 r Ω (1) n ˆV E n Ωn d 3 r Ω ( (0) n E n Ĥ0) Ωn or Tae, we have V n,n E (1) n δ d 3 r Ω n c (1) E (1) n V n,n c (1) (e n e n ) Ω n 0. and tae, we have V n,n d 3 r Ω ˆV n Ω n 0, for all. This last equation means operator ˆV is diagonal in the new basis. Therefore, the coefficients {b d } in Ω n M d1 b d Φ nd is chosen so that ˆV is diagonal. Example: Spin-orbit coupling (or LS-coupling). An electron moving in a central potential ˆV V (r) has Hamiltonian as Ĥ 0 h m + V (r).

12 6 CHAPTER 3. APPROXIMATION METHODS IN QM In general the eigenvalues E nl of Ĥ 0 are (l + 1)-fold degenerate (for hydrogen, E nl E n, and degeneracy is n -fold). One of relativistic effects (Dirac) is the addition of a potential, the so-called spin-orbit coupling (or LS-coupling) ˆV LS A (r) ˆL Ŝ. We can treat this ˆV LS as perturbation potential and write ˆV LS 1 A (r) (Ĵ ˆL Ŝ ), Ĵ ˆL + Ŝ As we now, the eigenstates of Ĥ0 can be represented by the product of orbital wavefunctions and spin wavefunctions, or nlm sm s nlm l sm s. Within the first-order degenerate PT, we need to use these nlm l sm s to diagonalize the perturbation operator ˆV LS and to obtain the zero-order wavefunction and first-order energy correction. We have already done so in the previous chapter. The zeroth-order wavefunction are angular momentum states nlsjm which are eigenstates of Ĵ and Ĵz with angular momentum quantum number, using s 1/ j l ± 1, m j, j + 1,, j 1, j. Hence, the energy level shift by LS-coupling is E (1) LS nlsjm ˆV [ LS nlsjm h j (j + 1) l (l + 1) 3 ] A (r) 4 nl. 3.3 Time-dependent perturbation theory Newton s equation As we have seen before, the time-dependent Schrödinger eq. is i h t Ψ (r, t) ĤΨ (r, t), i h t Ψ (r, t) ĤΨ (r, t). The expectation value of an arbitrary operator   (t), assumed Ψ (r, t) is normalized, Â Ψ Â Ψ d 3 r Ψ (r, t)  (t) Ψ (r, t) has the following evolution equation d  d 3 r dt t [ Ψ (r, t)  (t) Ψ (r, t)] d 3 r Ψ (r, t)  1 Ψ (r, t) + t i h d 3 r Ψ (r, t) ( ÂĤ ĤÂ) Ψ (r, t)

13 3.3. TIME-DEPENDENT PERTURBATION THEORY 63 or d dt   + 1 [Â, ] Ĥ. t i h Using this formula, Newton s equation becomes, using Ĥ ˆp /m + ˆV (r) d dt ˆp ˆV Time-dependent expansion Consider now the following Hamiltonian Ĥ Ĥ0 + ˆV (t) where Ĥ0 is time-independent with eigen eqs. Ĥ 0 Φ n (r) E n Φ n (r), n 1,, and the time dependent part of the Hamiltonian is ˆV (t). The general time-dependent wavefunction Ψ (r, t) can be expanded in the following general form as Ψ (r, t) n d n (t) Φ n (r) n c n (t) Φ n (r) e ient/ h and after substituting into the Schrödinger eq., we have ( E n c n + i h dc ) n Φ n (r) e ient/ h ( En + dt ˆV ) c n Φ n (r) e ient/ h n or n i h n dc n dt Φ n (r) e ie n t/ h n c n ˆV Φn (r) e ie n t/ h. Taing inner product with Φ n (r) on both sides of eq., we have the time-evolution equation for c n (t) i h dc n dt n V nn c n e iω nn t, n 1,, with V nn Φ n ˆV Φ n d 3 r Φ n ˆV Φ n, hω nn E n E n. these are a set of first-order, coupled equations for {c n, n 1,, }. Normally difficult to solve. For some special case, exact solution can be found. See Mandl, Quantum Mechanics, Section 9., P 198.

14 64 CHAPTER 3. APPROXIMATION METHODS IN QM Consider a typical initial condition: at t 0, the system is in one of the eigenstate, say, 1, namely c n (0) δ 1n. For ˆV (t) 0, the amplitudes c n (t) retain this initial value at all times and Ψ (r, t) n c n (t) Φ n (r) e ient/ h Φ 1 (r) e ie 1t/ h the time-dependent part is trivial Perturbation Theory Now we treat ˆV as perturbation potential and write the Hamiltonian as before and correspondingly c n c (0) n the evolution equation for c n becomes i h d dt Ĥ Ĥ0 + λ ˆV (t) + λc(1) n + λ c () n + ( c (0) n + λc (1) n + λ c () n + ) ( (0) c n + λc(1) n + λ c () n + ) λv nn e iω nn t. n Setting the coefficients of each power of λ equal on both sides we have a series successive eqs., i h d dt c(0) n 0, c (0) n (t) const. i h d dt c(1) n n c (0) n V nn eiω nn t i h d dt c() n n c (1) n V nn eiω nn t. Consider the typical initial condition: applying the perturbation ˆV (t) after t > 0, namely ˆV (t) { 0, t 0 0, t > 0 and assume for t 0, the system is in one of the eigenstates of Ĥ0, say 1, c n (0) c (0) n (t) δ 1n

15 3.3. TIME-DEPENDENT PERTURBATION THEORY 65 Hence the first-order equation becomes i h d dt c(1) n n c (0) n V nn eiω nn t V n1 e iω n1t Integrating these equations from t 0 to t > 0, subject to the condition that c n (0) δ 1n, we have c (1) 1 (t) i h c (1) n (t) 1 i h t t 0 ˆV 11 (t ) dt ˆV n1 (t ) e iω n1t dt, ω n1 E n E 1, n 1 h 0 where c (1) 1 (t) is the probability (in 1st-order approximation) that the system at t > 0 stays in the original state 1, and c (1) n (t) is the probability (in 1st-order approximation) that the system at t > 0 is in another state n (n 1). Example. A hydrogen atom is placed in a spatially homogeneous time-dependent electric field in the z-axis ε { 0, t 0 ε 0 e t/τ, t > 0 with τ > 0. At time t 0, the atom is in the (1s) ground state with wavefunction and energy φ 1s 1 e r/a 0, E 1s e, a 0 4πɛ 0 h. πa 3 8πɛ 0 a 0 me 0 Find the probability, in the 1st order perturbation theory, that after a sufficiently long time the atom is in the (p) excited state with wavefunction and energy φ p 1 r cos θ e r/a 0, 3πa 5 0 E p E 1s 4. Solution: According to the 1st-order perturbation theory, the probability amplitude for the transition (1s) (p) is with and c (t) 1 t ˆV p,1s (t ) e iω p,1st dt i h 0 ω p,1s E p E 1s h 3E 1s 4 h ˆV (t ) eε 0 e t /τ z eε 0 e t /τ r cos θ.

16 66 CHAPTER 3. APPROXIMATION METHODS IN QM The matrix element ˆV p,1s (t ) is ˆV p,1s (t ) p ˆV (t ) 1s eε 0 e t /τ p r cos θ 1s and the amplitude eε 0 e t /τ 1 r cos θ e r/a0 3πa 5 0 ( eε 0 e t /τ 1 4 πa πa 3 0 e r/a 0 ) ( π r 4 e 3r/a 0 dr cos θ sin θdθ 0 r sin θdrdθdφ ) ( π eε 0 e t /τ 1 4 4!5 a 5 0 πa π Ce t/τ, C 15/ 3 eε 0a 5 0 c (t) C i h t and the transition probability is 0 e iω p,1st t /τ dt C e iωp,1st t/τ 1 i h iω p,1s 1/τ c(t) C h 1 + e t/τ e t/τ cos(ω p,1s t) ω p,1s + 1/τ, and after a sufficiently long time, t, the probability is Fermi golden rule c ( ) (eε 0 a 0 ) (3E 1s /4) + ( h/τ). 0 ) dφ Now we consider the following simpler case, a step function perturbation potential ˆV (t) { 0, t 0 ˆV, t > 0 where ˆV is independent of time. What is the transition probability per unit time for the system originally in state 1 at t 0 to the state at t > 0? Solution: From the above 1sr-order perturbation theory, the time-integration in the transition probability amplitude can be carried out, c (1) 1 (t) 1 t V,1 (t ) e iω,1t dt V (,1 ) 1 e iω,1t i h 0 hω,1 V,1 ˆV 1, ω,1 E E 1 h

17 3.4. VARIATIONAL METHOD 67 and the probability is, using sin y (e iy e iy )/, P 1 (t) (1) c 1 (t) 4 V,1 sin (ω,1 t/) h πt ω,1 h V,1 D (ω,1, t) D (x, t) sin (xt/), x ω πt x,1 where function D (x, t) has a simple integral D (x, t) dx πt sin (xt/) x dx 1, using sin (ax) x dx πa, similar to the continuous delta function δ (x). In the limit t, the two become identical. So we have the transition probability given by, for large t P 1 (t) πt h and the probability per unit time This is the famous Fermi Golden rule. V,1 δ (ω,1 ) πt h V,1 δ (E E 1 ) P 1 (t) π h V,1 δ (E E 1 ). 3.4 Variational method Variational principle Variational principle (Rayleigh-Ritz): The expectation value of a given Hamiltonian Ĥ in any state Ψ is always greater or equal to the exact ground-state energy E 0, Ψ Ĥ Ψ Ψ Ψ E 0 the equality holds if Ψ is the exact ground state. Proof: Suppose Ĥ Φ n E n Φ n, n 0, 1,, E 0 < E n, for n > 0

18 68 CHAPTER 3. APPROXIMATION METHODS IN QM Write Ψ as a linear combination of Φ n Therefore and so Ψ C n Φ n C 0 Φ 0 + C n Φ n n0 n1 Ψ Ψ C 0 + C n n1 ( ) Ĥ Ψ Ĥ C 0 Φ 0 + C n Φ n n1 E 0 C 0 Φ 0 + C n E n Φ n n1 Ψ Ĥ Ψ E 0 C 0 + E n C n n1 > E 0 C 0 ) + E 0 C n E 0 ( C 0 + C n E 0 Ψ Ψ n1 n1 Ψ Ĥ Ψ E 0. Ψ Ψ In general, we do not now the exact ground state energy E 0. We wish to find an approximation for its value. By the variational principle, our approximation Ĥ using any trial wavefunction Ψ will always be greater than the exact value E 0. Hence, the lower the expectation value Ĥ (i.e., the closer to the exact value), the better the trial wavefunction. A typical varitaional calculation is as follows: we first choose a trial wavefunction Ψ with a few variational parameters α 1, α, and calculate the expectation value Ψ (α 1, α, ) E (α 1, α, ) Ψ Ĥ Ψ Ψ Ψ. Next, we minimize this E (α 1, α, ) through variational equations E (α 1, α, ) α 1 0 E (α 1, α, ) α 0

19 3.4. VARIATIONAL METHOD 69 The solution for (α 1, α, ) of these equations then put bac to E (α 1, α, ) and this minimum E represents the best estimate for the trial function of the form Ψ (α 1, α, ). Clearly, the success of this nonperturbative method will depend crucially on the form of the trial wave functions we choose to start with Examples Hydrogen atom. The trial function for the ground state of Hydrogen is choose as α Φ α Ce αr 3, C π the Hamiltonian is Ĥ h m e 4πɛ 0 r, and the expectation value is 1 ( r ) + 1 r r r r  (θφ) E α Φ α Ĥ Φ α h α m e α 4πɛ 0 hence E α α 0 α e m 4πɛ 0 h 1 a 0 where a 0 is the Bohr radius. So the best estimate for the ground state energy is and for the ground state is E α h m 1 a 0 e α 1 1 4πɛ 0 a 0 Φ 1 a 3 0π e r/a 0 e 4πɛ 0 a 0 both are actually exact. This is not surprising because the trial function has the correct form. Helium atom. In this case we do not now the exact result and we are dealing with two identical particles. Consider in general Helium-lie ions with the following Hamiltonian describing two electrons interacting each other and with a nucleus containing Z protons ( Ĥ h m 1 h m Ze ) + e 1. 4πɛ 0 r1 r 4πɛ 0 r 1

20 70 CHAPTER 3. APPROXIMATION METHODS IN QM We want to approximate the ground state of this system by a simple trial wavefunction (actually, PT), using the nowledge from hydrogen-lie ions. We now the eigenstates φ nlm (r) for a single electron Ĥ h m Ze 1 are 4πɛ 0 r Z 3 φ 1s (r) R 10 (r) Y 00 (θ, φ) e Zr/a 0 φ s (r) R 0 (r) Y 00 (θ, φ)... πa 3 0 Z 3 with φ 1s (r) as the ground state, and eigenvalues 8πa 3 0 ( 1 Zr ) e Zr/a 0 a 0 E nl E n e Z, n 1,,. 8πɛ 0 a 0 n We also now electrons have spin degree of freedom, indicated by two states or. We want to construct the simplest two-body wavefunction from these single-body state. There is a (anti)symmetry requirement we need to impose. The following is a trial wavefunction satisfying this requirement, Ψ 0 (1, ) φ 1s (r 1 ) φ 1s (r ) χ 1, χ 1 1 ( 1 1 ) hence Ψ (1, ) Ψ (, 1). This is a spin-singlet state. Another possible trial state is the spin-triplet, Ψ 0 (1, ) 1 [φ 1s (r 1 ) φ s (r ) φ s (r 1 ) φ 1s (r )] τ 1, τ 1 1, 1 ( ), 1 but this will give higher energy due to the present of the higher level state φ s (r 1 ). (It is quite easy to test experimentally if the ground state Helium is singlet or triplet.) Using the singlet state Ψ t (1, ), we evaluate the energy expectation value Ĥ h m 1 Ze 1 4πɛ 0 r 1 ( Z e ) + 5 Ze 8πɛ 0 a 0 4 8πɛ 0 a 0 ( Z + 5 ) 4 Z 5.50 Ry, Z + e 1 4πɛ 0 r 1 Ry, Ry e 8πɛ 0 a 0 which is only 6% above the experimental result of 5.81 Ry.

21 3.4. VARIATIONAL METHOD 71 Math Note: Evaluate 1 r 1 : 1 r 1 1 π ( Z a0 ) 6 d 3 r 1 d 3 r 1 1 r 1 r e Z(r 1+r )/a 0. First we use the expression (Fourier transformation) 1 r 1 r d 3 ei (r 1 r ) 4π (π) 3 hence now and finally 1 r 1 1 π ( Z a0 ) 6 d 3 r 1 e i r 1 Zr 1 /a 0 1 r 1 4Z πa 0 0 d 3 4π (π) 3 d 3 r 1 e i r 1 Zr 1 /a 0 16πZ/a 0 [ + (Z/a 0 ) ] dx (x + 1) 4 5 Z. 8 a 0

22 7 CHAPTER 3. APPROXIMATION METHODS IN QM We can improve our wavefunction by introducing a variational parameter Z Z 3 φ 1s (r) e Z r/a 0, Ψ πa 3 0 (1, ) φ 1s (r 1 ) φ 1s (r ) χ 1 0 and repeat our calculation as before, see example one, Ĥ h Z m 1 Ze 1 + e 1 4πɛ 0 r 1 4πɛ Z 0 r 1 Z ( Z e ZZ e ) + 5 Z e 8πɛ 0 a 0 4πɛ 0 a 0 4 8πɛ 0 a 0 (Z Z Z + 5 ) 8 Z Ry, Ry e 8πɛ 0 a 0 and variational eq. Ĥ Z Z 0 Z Z 5 16 and the estimated ground-state energy Ĥ Z Z 5/16 ( Z 16) 5 Ry, Z 7 16 Ry 5.70 Ry which is closer to the experimental result of 5.81 Ry. (higher no more than %). Physically, the variational parameter Z represents the mutual screening by the two electrons. Now how does one go from here? Can we further improve our estimate and how? A more sophisticated variational wavefunction (Hylleraas) is Ψ (1, ) P (s, t, u) Ψ 0 (1, ) s Z a 0 (r 1 + r ), t Z a 0 (r 1 r ), u Z a 0 r 1 where P (s, t, u) is expanded in power series of s, t and u P (s, t, u) c n,l,m s n t l u m l,n,m and variational eqs. are c n,l,m Ĥ 0, Z Ĥ 0

23 3.4. VARIATIONAL METHOD 73 and by including 10 c n,l,m parameters, the accuracy can reach to 6-7 digits. See Intermediate QM, 1964, H.A. Bethe. Variational method and diagonalization. For a given basis set (truncated or untruncated), { n, n 1,,, N}, the trial wavefunction can be written as N N Ψ c n n, Ψ n c n n1 n1 where {c n, c n; n 1,,, N} are variational parameters determined by or or δ δc n Ĥ δ δc n Ĥ 0, Ĥ Ψ Ĥ Ψ Ψ Ψ Ψ Ψ N n 1 H nn c n Ψ Ĥ Ψ c n Ψ Ψ 0, H nn n Ĥ n N H nn c n Ĥ c n, n 1,,, N. n 1 This is precisely the eigenequation for the Hamiltonian matrix {H nn } with the energy eigenvalue Ĥ. Therefore, the variational method and diagonalization is equivalent.

Quantum Mechanics Solutions

Quantum Mechanics Solutions Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H

More information

PHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.

PHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m. PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms

More information

Further Quantum Mechanics Problem Set

Further Quantum Mechanics Problem Set CWPP 212 Further Quantum Mechanics Problem Set 1 Further Quantum Mechanics Christopher Palmer 212 Problem Set There are three problem sets, suitable for use at the end of Hilary Term, beginning of Trinity

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 14, 015 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this

More information

16.1. PROBLEM SET I 197

16.1. PROBLEM SET I 197 6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,

More information

(Again, this quantity is the correlation function of the two spins.) With z chosen along ˆn 1, this quantity is easily computed (exercise):

(Again, this quantity is the correlation function of the two spins.) With z chosen along ˆn 1, this quantity is easily computed (exercise): Lecture 30 Relevant sections in text: 3.9, 5.1 Bell s theorem (cont.) Assuming suitable hidden variables coupled with an assumption of locality to determine the spin observables with certainty we found

More information

PHY413 Quantum Mechanics B Duration: 2 hours 30 minutes

PHY413 Quantum Mechanics B Duration: 2 hours 30 minutes BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO

More information

The 3 dimensional Schrödinger Equation

The 3 dimensional Schrödinger Equation Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum

More information

Lectures 21 and 22: Hydrogen Atom. 1 The Hydrogen Atom 1. 2 Hydrogen atom spectrum 4

Lectures 21 and 22: Hydrogen Atom. 1 The Hydrogen Atom 1. 2 Hydrogen atom spectrum 4 Lectures and : Hydrogen Atom B. Zwiebach May 4, 06 Contents The Hydrogen Atom Hydrogen atom spectrum 4 The Hydrogen Atom Our goal here is to show that the two-body quantum mechanical problem of the hydrogen

More information

Qualification Exam: Quantum Mechanics

Qualification Exam: Quantum Mechanics Qualification Exam: Quantum Mechanics Name:, QEID#76977605: October, 2017 Qualification Exam QEID#76977605 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1/2 particles

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 10, 2018 10:00AM to 12:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of

More information

7.1 Variational Principle

7.1 Variational Principle 7.1 Variational Principle Suppose that you want to determine the ground-state energy E g for a system described by H, but you are unable to solve the time-independent Schrödinger equation. It is possible

More information

Lecture 8: Radial Distribution Function, Electron Spin, Helium Atom

Lecture 8: Radial Distribution Function, Electron Spin, Helium Atom Lecture 8: Radial Distribution Function, Electron Spin, Helium Atom Radial Distribution Function The interpretation of the square of the wavefunction is the probability density at r, θ, φ. This function

More information

Schrödinger equation for central potentials

Schrödinger equation for central potentials Chapter 2 Schrödinger equation for central potentials In this chapter we will extend the concepts and methods introduced in the previous chapter for a one-dimensional problem to a specific and very important

More information

The Central Force Problem: Hydrogen Atom

The Central Force Problem: Hydrogen Atom The Central Force Problem: Hydrogen Atom B. Ramachandran Separation of Variables The Schrödinger equation for an atomic system with Z protons in the nucleus and one electron outside is h µ Ze ψ = Eψ, r

More information

Lecture 4 Quantum mechanics in more than one-dimension

Lecture 4 Quantum mechanics in more than one-dimension Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts

More information

r 2 dr h2 α = 8m2 q 4 Substituting we find that variational estimate for the energy is m e q 4 E G = 4

r 2 dr h2 α = 8m2 q 4 Substituting we find that variational estimate for the energy is m e q 4 E G = 4 Variational calculations for Hydrogen and Helium Recall the variational principle See Chapter 16 of the textbook The variational theorem states that for a Hermitian operator H with the smallest eigenvalue

More information

Lecture 4 Quantum mechanics in more than one-dimension

Lecture 4 Quantum mechanics in more than one-dimension Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts

More information

0 + E (1) and the first correction to the ground state energy is given by

0 + E (1) and the first correction to the ground state energy is given by 1 Problem set 9 Handout: 1/24 Due date: 1/31 Problem 1 Prove that the energy to first order for the lowest-energy state of a perturbed system is an upper bound for the exact energy of the lowest-energy

More information

Chapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)

Chapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1) Tor Kjellsson Stockholm University Chapter 6 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: where α is a constant. H = αδ(x a/ ( a Find the first-order correction

More information

Lecture 12. The harmonic oscillator

Lecture 12. The harmonic oscillator Lecture 12 The harmonic oscillator 107 108 LECTURE 12. THE HARMONIC OSCILLATOR 12.1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the time-independent

More information

1 Rayleigh-Schrödinger Perturbation Theory

1 Rayleigh-Schrödinger Perturbation Theory 1 Rayleigh-Schrödinger Perturbation Theory All perturbative techniques depend upon a few simple assumptions. The first of these is that we have a mathematical expression for a physical quantity for which

More information

Lecture 15 From molecules to solids

Lecture 15 From molecules to solids Lecture 15 From molecules to solids Background In the last two lectures, we explored quantum mechanics of multi-electron atoms the subject of atomic physics. In this lecture, we will explore how these

More information

Physics 115C Homework 3

Physics 115C Homework 3 Physics 115C Homework 3 Problem 1 In this problem, it will be convenient to introduce the Einstein summation convention. Note that we can write S = i S i i where the sum is over i = x,y,z. In the Einstein

More information

Quantum Mechanics I Physics 5701

Quantum Mechanics I Physics 5701 Quantum Mechanics I Physics 5701 Z. E. Meziani 02/23//2017 Physics 5701 Lecture Outline 1 General Formulation of Quantum Mechanics 2 Measurement of physical quantities and observables 3 Representations

More information

C. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.

C. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0. Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is

More information

Angular Momentum - set 1

Angular Momentum - set 1 Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x ] =

More information

Printed from file Manuscripts/Stark/stark.tex on February 24, 2014 Stark Effect

Printed from file Manuscripts/Stark/stark.tex on February 24, 2014 Stark Effect Printed from file Manuscripts/Stark/stark.tex on February 4, 4 Stark Effect Robert Gilmore Physics Department, Drexel University, Philadelphia, PA 94 February 4, 4 Abstract An external electric field E

More information

1 Position Representation of Quantum State Function

1 Position Representation of Quantum State Function C/CS/Phys C191 Quantum Mechanics in a Nutshell II 10/09/07 Fall 2007 Lecture 13 1 Position Representation of Quantum State Function We will motivate this using the framework of measurements. Consider first

More information

Intermission: Let s review the essentials of the Helium Atom

Intermission: Let s review the essentials of the Helium Atom PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the

More information

Chapter 29. Quantum Chaos

Chapter 29. Quantum Chaos Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical

More information

The Hydrogen Atom Chapter 20

The Hydrogen Atom Chapter 20 4/4/17 Quantum mechanical treatment of the H atom: Model; The Hydrogen Atom Chapter 1 r -1 Electron moving aroundpositively charged nucleus in a Coulombic field from the nucleus. Potential energy term

More information

Consequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators

Consequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators VI. SPIN-ADAPTED CONFIGURATIONS A. Preliminary Considerations We have described the spin of a single electron by the two spin functions α(ω) α and β(ω) β. In this Sect. we will discuss spin in more detail

More information

Time dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012

Time dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012 Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system

More information

SCIENCE VISION INSTITUTE For CSIR NET/JRF, GATE, JEST, TIFR & IIT-JAM Web:

SCIENCE VISION INSTITUTE For CSIR NET/JRF, GATE, JEST, TIFR & IIT-JAM Web: Test Series: CSIR NET/JRF Exam Physical Sciences Test Paper: Quantum Mechanics-I Instructions: 1. Attempt all Questions. Max Marks: 185 2. There is a negative marking of 1/4 for each wrong answer. 3. Each

More information

Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom

Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Example / Worked Problems What is the ratio of the

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 12, 2011 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this

More information

Preliminary Quantum Questions

Preliminary Quantum Questions Preliminary Quantum Questions Thomas Ouldridge October 01 1. Certain quantities that appear in the theory of hydrogen have wider application in atomic physics: the Bohr radius a 0, the Rydberg constant

More information

Perturbation Theory. D. Rubin. December 2, Stationary state perturbation theory

Perturbation Theory. D. Rubin. December 2, Stationary state perturbation theory Perturbation Theory D. Rubin December, 010 Lecture 3-41 November 10- December 3, 010 1 Stationary state perturbation theory 1.1 Nondegenerate Formalism We have a Hamiltonian H = H 0 + V and we suppose

More information

charges q r p = q 2mc 2mc L (1.4) ptles µ e = g e

charges q r p = q 2mc 2mc L (1.4) ptles µ e = g e APAS 5110. Atomic and Molecular Processes. Fall 2013. 1. Magnetic Moment Classically, the magnetic moment µ of a system of charges q at positions r moving with velocities v is µ = 1 qr v. (1.1) 2c charges

More information

Lecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators

Lecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform

More information

PHYS-454 The position and momentum representations

PHYS-454 The position and momentum representations PHYS-454 The position and momentum representations 1 Τhe continuous spectrum-a n So far we have seen problems where the involved operators have a discrete spectrum of eigenfunctions and eigenvalues.! n

More information

Sample Quantum Chemistry Exam 2 Solutions

Sample Quantum Chemistry Exam 2 Solutions Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)

More information

Single Electron Atoms

Single Electron Atoms Single Electron Atoms In this section we study the spectrum and wave functions of single electron atoms. These are hydrogen, singly ionized He, doubly ionized Li, etc. We will write the formulae for hydrogen

More information

1 Recall what is Spin

1 Recall what is Spin C/CS/Phys C191 Spin measurement, initialization, manipulation by precession10/07/08 Fall 2008 Lecture 10 1 Recall what is Spin Elementary particles and composite particles carry an intrinsic angular momentum

More information

Physical Chemistry Quantum Mechanics, Spectroscopy, and Molecular Interactions. Solutions Manual. by Andrew Cooksy

Physical Chemistry Quantum Mechanics, Spectroscopy, and Molecular Interactions. Solutions Manual. by Andrew Cooksy Physical Chemistry Quantum Mechanics, Spectroscopy, and Molecular Interactions Solutions Manual by Andrew Cooksy February 4, 2014 Contents Contents i Objectives Review Questions 1 Chapter Problems 11 Notes

More information

Problem 1: Spin 1 2. particles (10 points)

Problem 1: Spin 1 2. particles (10 points) Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a

More information

3: Many electrons. Orbital symmetries. l =2 1. m l

3: Many electrons. Orbital symmetries. l =2 1. m l 3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience

More information

The Quantum Heisenberg Ferromagnet

The Quantum Heisenberg Ferromagnet The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,

More information

Physics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I

Physics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from

More information

A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians

A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians Ali Mostafazadeh Department of Mathematics, Koç University, Istinye 886, Istanbul, TURKEY Abstract For a T -periodic

More information

Angular Momentum - set 1

Angular Momentum - set 1 Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x = 0,

More information

2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements

2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements 1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical

More information

Physics 4022 Notes on Density Matrices

Physics 4022 Notes on Density Matrices Physics 40 Notes on Density Matrices Definition: For a system in a definite normalized state ψ > the density matrix ρ is ρ = ψ >< ψ 1) From Eq 1 it is obvious that in the basis defined by ψ > and other

More information

LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (07/2017) =!2 π 2 a cos π x

LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (07/2017) =!2 π 2 a cos π x LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (7/17) 1. For a particle trapped in the potential V(x) = for a x a and V(x) = otherwise, the ground state energy and eigenfunction

More information

Harmonic Oscillator I

Harmonic Oscillator I Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering

More information

The Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case:

The Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about

More information

Solutions to chapter 4 problems

Solutions to chapter 4 problems Chapter 9 Solutions to chapter 4 problems Solution to Exercise 47 For example, the x component of the angular momentum is defined as ˆL x ŷˆp z ẑ ˆp y The position and momentum observables are Hermitian;

More information

Chapter 2 Heisenberg s Matrix Mechanics

Chapter 2 Heisenberg s Matrix Mechanics Chapter 2 Heisenberg s Matrix Mechanics Abstract The quantum selection rule and its generalizations are capable of predicting energies of the stationary orbits; however they should be obtained in a more

More information

Physics 401: Quantum Mechanics I Chapter 4

Physics 401: Quantum Mechanics I Chapter 4 Physics 401: Quantum Mechanics I Chapter 4 Are you here today? A. Yes B. No C. After than midterm? 3-D Schroedinger Equation The ground state energy of the particle in a 3D box is ( 1 2 +1 2 +1 2 ) π2

More information

Physics 221B Spring 2018 Notes 34 The Photoelectric Effect

Physics 221B Spring 2018 Notes 34 The Photoelectric Effect Copyright c 2018 by Robert G. Littlejohn Physics 221B Spring 2018 Notes 34 The Photoelectric Effect 1. Introduction In these notes we consider the ejection of an atomic electron by an incident photon,

More information

Angular momentum. Quantum mechanics. Orbital angular momentum

Angular momentum. Quantum mechanics. Orbital angular momentum Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular

More information

Quantum Physics Lecture 9

Quantum Physics Lecture 9 Quantum Physics Lecture 9 Potential barriers and tunnelling Examples E < U o Scanning Tunelling Microscope E > U o Ramsauer-Townsend Effect Angular Momentum - Orbital - Spin Pauli exclusion principle potential

More information

The Particle in a Box

The Particle in a Box Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:

More information

Total Angular Momentum for Hydrogen

Total Angular Momentum for Hydrogen Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p

More information

Quantum Mechanics. L. Del Debbio, University of Edinburgh Version 0.9

Quantum Mechanics. L. Del Debbio, University of Edinburgh Version 0.9 Quantum Mechanics L. Del Debbio, University of Edinburgh Version 0.9 November 26, 2010 Contents 1 Quantum states 1 1.1 Introduction..................................... 2 1.1.1 Experiment with classical

More information

Waves and the Schroedinger Equation

Waves and the Schroedinger Equation Waves and the Schroedinger Equation 5 april 010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form

More information

Introduction to Vibrational Spectroscopy

Introduction to Vibrational Spectroscopy Introduction to Vibrational Spectroscopy Harmonic oscillators The classical harmonic oscillator The uantum mechanical harmonic oscillator Harmonic approximations in molecular vibrations Vibrational spectroscopy

More information

Linear Algebra in Hilbert Space

Linear Algebra in Hilbert Space Physics 342 Lecture 16 Linear Algebra in Hilbert Space Lecture 16 Physics 342 Quantum Mechanics I Monday, March 1st, 2010 We have seen the importance of the plane wave solutions to the potentialfree Schrödinger

More information

Physics 221A Fall 2017 Notes 25 The Zeeman Effect in Hydrogen and Alkali Atoms

Physics 221A Fall 2017 Notes 25 The Zeeman Effect in Hydrogen and Alkali Atoms Copyright c 2017 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 25 The Zeeman Effect in Hydrogen and Alkali Atoms 1. Introduction The Zeeman effect concerns the interaction of atomic systems with

More information

The Electronic Structure of Atoms

The Electronic Structure of Atoms The Electronic Structure of Atoms Classical Hydrogen-like atoms: Atomic Scale: 10-10 m or 1 Å + - Proton mass : Electron mass 1836 : 1 Problems with classical interpretation: - Should not be stable (electron

More information

Chem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals

Chem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals Chem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals Pre-Quantum Atomic Structure The existence of atoms and molecules had long been theorized, but never rigorously proven until the late 19

More information

Further Quantum Physics. Hilary Term 2009 Problems

Further Quantum Physics. Hilary Term 2009 Problems Further Quantum Physics John Wheater Hilary Term 009 Problems These problem are labelled according their difficulty. So some of the problems have a double dagger to indicate that they are a bit more challenging.

More information

5.111 Lecture Summary #6

5.111 Lecture Summary #6 5.111 Lecture Summary #6 Readings for today: Section 1.9 (1.8 in 3 rd ed) Atomic Orbitals. Read for Lecture #7: Section 1.10 (1.9 in 3 rd ed) Electron Spin, Section 1.11 (1.10 in 3 rd ed) The Electronic

More information

Collection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators

Collection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators Basic Formulas 17-1-1 Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum

More information

If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.

If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk

More information

H atom solution. 1 Introduction 2. 2 Coordinate system 2. 3 Variable separation 4

H atom solution. 1 Introduction 2. 2 Coordinate system 2. 3 Variable separation 4 H atom solution Contents 1 Introduction 2 2 Coordinate system 2 3 Variable separation 4 4 Wavefunction solutions 6 4.1 Solution for Φ........................... 6 4.2 Solution for Θ...........................

More information

The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field:

The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field: SPIN 1/2 PARTICLE Stern-Gerlach experiment The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field: A silver atom

More information

Quantum Theory of Angular Momentum and Atomic Structure

Quantum Theory of Angular Momentum and Atomic Structure Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the

More information

221B Lecture Notes Scattering Theory II

221B Lecture Notes Scattering Theory II 22B Lecture Notes Scattering Theory II Born Approximation Lippmann Schwinger equation ψ = φ + V ψ, () E H 0 + iɛ is an exact equation for the scattering problem, but it still is an equation to be solved

More information

arxiv: v1 [quant-ph] 25 Apr 2008

arxiv: v1 [quant-ph] 25 Apr 2008 Generalized Geometrical Phase in the Case of Continuous Spectra M. Maamache and Y. Saadi Laboratoire de Physique Quantique et Systèmes Dynamiques, Faculté des Sciences,Université Ferhat Abbas de Sétif,

More information

Physics 741 Graduate Quantum Mechanics 1 Solutions to Midterm Exam, Fall x i x dx i x i x x i x dx

Physics 741 Graduate Quantum Mechanics 1 Solutions to Midterm Exam, Fall x i x dx i x i x x i x dx Physics 74 Graduate Quantum Mechanics Solutions to Midterm Exam, Fall 4. [ points] Consider the wave function x Nexp x ix (a) [6] What is the correct normaliation N? The normaliation condition is. exp,

More information

Angular momentum and spin

Angular momentum and spin Luleå tekniska universitet Avdelningen för Fysik, 007 Hans Weber Angular momentum and spin Angular momentum is a measure of how much rotation there is in particle or in a rigid body. In quantum mechanics

More information

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

More information

SECOND QUANTIZATION PART I

SECOND QUANTIZATION PART I PART I SECOND QUANTIZATION 1 Elementary quantum mechanics We assume that the reader is already acquainted with elementary quantum mechanics. An introductory course in quantum mechanics usually addresses

More information

Exam in TFY4205 Quantum Mechanics Saturday June 10, :00 13:00

Exam in TFY4205 Quantum Mechanics Saturday June 10, :00 13:00 NTNU Page 1 of 9 Institutt for fysikk Contact during the exam: Professor Arne Brataas Telephone: 7359367 Exam in TFY5 Quantum Mechanics Saturday June 1, 6 9: 13: Allowed help: Alternativ C Approved Calculator.

More information

8.1 The hydrogen atom solutions

8.1 The hydrogen atom solutions 8.1 The hydrogen atom solutions Slides: Video 8.1.1 Separating for the radial equation Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4 (up to Solution of the hydrogen radial

More information

eff (r) which contains the influence of angular momentum. On the left is

eff (r) which contains the influence of angular momentum. On the left is 1 Fig. 13.1. The radial eigenfunctions R nl (r) of bound states in a square-well potential for three angular-momentum values, l = 0, 1, 2, are shown as continuous lines in the left column. The form V (r)

More information

Physics 443, Solutions to PS 2

Physics 443, Solutions to PS 2 . Griffiths.. Physics 443, Solutions to PS The raising and lowering operators are a ± mω ( iˆp + mωˆx) where ˆp and ˆx are momentum and position operators. Then ˆx mω (a + + a ) mω ˆp i (a + a ) The expectation

More information

Fermionic Algebra and Fock Space

Fermionic Algebra and Fock Space Fermionic Algebra and Fock Space Earlier in class we saw how harmonic-oscillator-like bosonic commutation relations [â α,â β ] = 0, [ ] â α,â β = 0, [ ] â α,â β = δ α,β (1) give rise to the bosonic Fock

More information

Chemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer.

Chemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer. B Chemistry 3502/4502 Exam III All Hallows Eve/Samhain, 2003 1) This is a multiple choice exam. Circle the correct answer. 2) There is one correct answer to every problem. There is no partial credit. 3)

More information

Quantum Mechanics Final Exam Solutions Fall 2015

Quantum Mechanics Final Exam Solutions Fall 2015 171.303 Quantum Mechanics Final Exam Solutions Fall 015 Problem 1 (a) For convenience, let θ be a real number between 0 and π so that a sinθ and b cosθ, which is possible since a +b 1. Then the operator

More information

Physics 70007, Fall 2009 Answers to Final Exam

Physics 70007, Fall 2009 Answers to Final Exam Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,

More information

Quantum Mechanics in Three Dimensions

Quantum Mechanics in Three Dimensions Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent

More information

For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as

For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as VARIATIO METHOD For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as Variation Method Perturbation Theory Combination

More information

The Spin (continued). February 8, 2012

The Spin (continued). February 8, 2012 The Spin continued. Magnetic moment of an electron Particle wave functions including spin Stern-Gerlach experiment February 8, 2012 1 Magnetic moment of an electron. The coordinates of a particle include

More information

Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,

Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you

More information

Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures

Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures Jeong Won Kang Department of Chemical Engineering Korea University Subjects Three Basic Types of Motions

More information

POEM: Physics of Emergent Materials

POEM: Physics of Emergent Materials POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Reference: Bernevig Topological Insulators and Topological Superconductors Tutorials:

More information

Lecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics

Lecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of

More information