量子力学 Quantum mechanics. School of Physics and Information Technology
|
|
- Nathan Carpenter
- 5 years ago
- Views:
Transcription
1 量子力学 Quantum mechanics School of Physics and Information Technology Shaanxi Normal University
2 Chapter 9 Time-dependent perturation theory
3 Chapter 9 Time-dependent perturation theory 9.1 Two-level systems 9.2 Emission and asorption of radiation 9.3 Spontaneous emission
4 9.1 Two-level systems 9.2 Emission and asorption of radiation 9.3 Spontaneous emission
5 9.1 Two-level systems Suppose that there are just two states of the system. They are eigenstates of the unpertured Hamiltonian,and they are eigenstates orthonormal: H ψ = Eψ and H ψ = Eψ a a a ψ ψ = δ a a Any state can e expressed as a linear comination of them () = ca a + c ψ ψ ψ In the asence of any perturation,each component evolves with its characteristic exponential factor: ψ ( t) = c ψ e + c ψ e ieat / h ieat / h a a 2 2 c + c = 1 a
6 9.1.1 The pertured system Now suppose we turn on a time-dependent perturation, then the wave function can expressed as follows: ψ ( t) = c ( t) ψ e + c ( t) ψ e ieat / h ieat / h a a If the particle started out in the state ψ a We solve for c ( t) and c ( t) y demanding that ψ (t) satisfy the tim-dependent a SchrÖdinger equation Hψ = So, we find that ih ψ, t where H = H + H ( t)
7 c [ H ψ ] e + c [ H ψ ] e + c [ H ψ ] e ieat / h iet / h ieat / h a a a a + c [ H ψ ] e = ih[ c& ψ e + c& ψ e iet / h ieat / h iet / h a a ie ie + c ψ e + c ψ e h h a ieat / h iet / h a a The first two terms on the left cancel the last two terms on the right, and hence [ ψ ] ie a t / h + [ ψ ] ie t / h a a c H e c H e ψ ie a t / h & ψ ie t / h a a = ih[ c& e + c e ] Take the inner product with ψ, and exploit the orthogonality of ψ and ψ a a ]
8 For short, we define H ψ H ψ ij i j * Note that the Hermiticity of H entails H ji = ( H ji ). M ultiplying - i h e ieat / h through y ( / ), we conclude that i i( E Ea ) t / h c & a = [ cahaa + chae ] h Similarly, the inner product with ψ picks out c& : i( E Ea ) t / c i [ ch cahae h & = + ] h Typically, the diagonal matrix elements of H vanish: H aa = H =
9 In that case the equations simplify where & i iω t a a c = H e c & h i iω t a c = H e c ω h E E h a (1) Well assume that E E, so ω a
10 9.1.2 Time-Dependent Perturation Theory So far, everything is exact: we have made no assumption aout the size of the perturation. But if H is "small", we can solve Equation (1) y a process of successive approximations, as follows. Suppose the particle starts out in the lower state: c a Zeroth Order: () = 1, c () =. If there were no perturation at all, they would stay this way foreever c ( t) = 1, c =. ( ) ( ) a To calculate the first-order approximation, we insert these values on the right side of Equation (1)
11 First Order: dca (1) = ca = 1; dt dc t i iωt (1) i iωt = Hae c = H ( ) a t e dt dt h h Now we insert these expressions on the right to otain the second-order approximation: Second-Order: dc t a i H e iωt H i t ( t ) e ω = dt ( i ) a h a dt h 1 t c t H t e ω H t e dt dt h t (2) iω t i t a ( ) = 1 ( ) ( ) 2 a a
12 Notice that in my notation Notes: c (2) a ( t) includes the zeroth order term; the second-order correction would e the integral term alone. In principle, we could continue this ritual indefinitely, always inserting the n th order approximation into the right side of Equation (1) and solving for the ( n+ 1) order. Notice that c is modified in every even order, and c in every odd order. a th
13 9.1.3 Sinusoidal Perturation Suppose the perturation has sinusoidal time dependence: v v H ( r, t) = V ( r ) cos( ωt) so that where a To first order we have H = V cos( ωt), V = ψ a V ψ a a i t iωt dt c ( t) V a cos( ωt ) e h iv t a = - e + e dt 2h i( ω + ω ) t i( ω ω ) t i( ω + ω ) t i( ω ω ) t V a e 1 e 1 = h ω + ω ω ω This is the answer, ut it is a little cumersome to work with.
14 Dropping the first term: We assume ω + ω>> ω ω. so i( ω ω) t/2 Va e c ( t) e e 2h ω ω V = i h a sin[( ω ω) t/2] e ω ω i( ω ω) t/2 i( ω ω) t/2 i( ω ω) t/2 The transition proa ility the proa ility that a particle which started out in the state ψ will e found, at time, in the state ψ : V sin [( ω ω) t/2] P ( t) = c ( t) (2) a a a 2 2 h ( ω ω)
15 As a function of time, the transition proaility oscillates sinusoidally. P(t) V a h( ω ω ) 2 2π ω ω 4π ω ω ω 6π ω t Fig. 1 Transition proaility as a function of time, for a sinusoidal perturation
16 The proaility of a transition is greatest when the driving frequency is close to the natural frequency P(t) ( ω 2 πt) ω ( ω + 2 πt) ω Fig. 2 Transition proaility as a function of driving frequency
17 9.2 Emission and asorption of radiation Electromagnetic Waves An atom, in the presence of a passing light wave, responds primarily ot the electic component. If the wavelenght is long, we can ignore the spatial variation in the field; the atom is exposed to a sinusoidally oscillating electric field v E = E cos( ωt) kˆ The perturing Hamiltonian is H = qe z cos( ωt) where q is the charge of the electron. Evidently H = E cos( ωt), where = q ψ z ψ a a
18 Chapter 9 TIME-DEPENDENT PERTURBATION THEORY 9.1 Two-level systems 9.2 Emission and asorption of radiation 9.3 Spontaneous emission
19 An electromagnetic wave consists of transverse oscillating electric and magnetic fields. Electric field H v E v o An electromagnetic wave u v x Magnetic field
20 Typically, ψ is an even or odd function of z; in 2 either case z is odd, and integrates to zero. This licenses our usual assumption that the diagonal ψ matrix elements of H vanish. Thus the interaction of light with matter is governed y precisely the kind of oscillatory perturation with Va = E
21 9.2.2 Asorption, Stimulated Emission, and Spontaneous Emission If an atom starts out in the "lower" state ψ a, and you shine a polarized monochromatic eam of light on it, the proaility of a transition to the "upper" state is given y Equation (2), which takes the form P a ( t) 2 2 E ω ω 2 2 ( ω ω ) sin [( ) t / 2] = h 2 2 E sin [( ω ω) t / 2] = h ( ω ω) ψ Of course, the proaility of a transition down to to the lower level: P a ( t)
22 Three ways in which light interacts with atoms: (a) Asorption In this process, the atom asors energy E - E a = hω from the electromagnetic field. we say that it has "asored a photon" (a) Asorption
23 () Stimulated emission If the particle is in the upper state, and you shine light on it, it can make a transition ot the lower state, and in fact the proaility of such a transition is exactly the same as for a transition upward from the lower state. which process, which was first discovered y Einstein, is called stimulated emission. () Stimulated emission
24 (c) Spontaneous emission If an atom in the excited state makes a transition downward, with the release of a photon ut without any applied electromagnetic field to initiate the process. Thi s process is called spontane ous emission. (c) Spontaneous emission
25 9.2.3 Incoherent Perturation The energy density in an electromagnetic wave is ε 2 u = E 2 where E is the amplitude of the electric field. So the transition proaility is proportional portional to the energy density of the fields: P a 2u sin [( ω ω) t / 2] ( t) = ε ω ω h ( ) This is for monochromatic perturation, consisting of a single frequency ω.
26 In many applications the system is exposed to electromagnetic waves at a whole range of frequencies; in that case u ρ( ω) dω, and the net transition proality takes the form of an integral: P ( ) a t = 2u ε 2 2 ρ( ω) 2 2 h ( ω ω) sin [( ω ω) t / 2] dω replace ρ ( ω ) y ρ ( ω ) and take it outside the integral, we get 2 2 sin [( ω ω) t / 2] ρ ω 2 2 ( ω ω) 2 Pa ( t) ( ) dω ε h Changing variales to x ( ω ω) / 2, extending the limits of integration to x = ±, and looking up the define integral sin x x dx = π
27 we find 2 P t t 2 ( ) ( ) a ρ ω 2 εh This time the transition proaility is proportional ot t. When we hit the system with an incoherent spread of frequencies. The transition rate is Note : now 2 R = π a ρ( ω 2 ) εh the perturing wave is coming in along the x-direction and polarized in the z-directi on. a constant :
28 But we shall e interested in the case of an atom athed in radiation coming from all direction, and with all possile polarizations. What we 2 need, in place of,is th e average of where q ψ v r ψ and the average is over oth polarizations nˆ ( nˆ ) 2, and over all incident direction. This averaging can e carried out as follows: a
29 Polarization: For propagation in the z-direction, the two possile polarizations are iˆ and ˆj, so the polarization average is (ˆ n ) p = [( i ˆ ) + ( ˆ j ) ] = ( x y) = sin θ 2 2 where θ is the angle etween and the direction of propagation.
30 Propagation direction: Now lets set polar axis along and integrate over all propagation directions to get the polarization-propagation average: ( n ˆ ) = sin θ sinθdθdφ pp 4π 2 = sin 2 2 π 3 θdθ = 4 3 So the transition rate for stimulated emission from state to state a, under the influence of incoherent, unpolarized light incident from all directions, is
31 So the transition rate for stimulated emission from state to state a, under the influence of incoherent, unpolarized light incident from al l directions, is = π a ρ( ω ) ε h R a Where is the matrix element of the electric dipole moment etween the two states and ρ( ω ) is the energy density in the fields, per unit frequency, evaluated at ω = ( E E )/ h. a
32 Chapter 9 TIME-DEPENDENT PERTURBATION THEORY 9.1 Two-level systems 9.2 Emission and asorption of radiation 9.3 Spontaneous emission
33 9.3 Spontaneous emission Einsteins A and B coefficients Picture a container of atoms, N of them in the lower state ( ψ ), and N of them in the upper state ( ψ ). a Let A e the spontaneous emission rate, so that the numer of particle leaving the upper state y this process, per unit time, is N A. The transition rate for stimulated emission is proportional to the energy density of the electromagnetic field---call it B ρ ( ω ). The numer of particles leaving the upper state y this mechanism, per unit time, is N B ρ ( ω ). a a a
34 The asorption rate is likewise proportional to ρ( ω )---call it B ρ( ω ); a The numer of particles per unit joining the upper level is therefore N B ρ( ω a a ). All told, then, dn = N A NB ρ( ω a ) + NaB ρ( ω a ) dt Suppose that these atoms are in thermal equilirium with the amient field, so that the numer of particle in each level is constant.
35 In that case dn / dt =, ρ ( ω ) = and it follows that A ( N / N ) B B a a a The numer of particles with energy E, in thermal equilirium at temperature the Boltzmann factor, so N N and hence a ρ ( ω ) Ea / kbt e = = Ea / kbt e = hω / k T e B A B e a hω / k T T, is proportional to B B a
36 But Plancks lackody formula tells us the energy density of thermal radiation: ρ ( ω ) = h π hω / kt c e B a Comparing the two expressions, we conclude that and ω 3 ω h Ba = Ba and A = 2 3 π c π 2 Ba = 2 3ε h and it follows that the spontaneous emission rateis A = 3 2 ω 3πε hc 3 B a
37 9.3.2 The Lifetime of an Excited State Suppose, now, that you have a ottle full of atoms, with N ( t), of them in the excited state. As a result of spontaneous emission, this numer w ill d ecrease as time goes on; specifically, in a time interval dt you will lose a fraction of them: Solving for N dn = AN dt ( t), we find N ( t) = N () e At Adt
38 evidently the numer remaining in the excited state decreases exponentially, with a time constant 1 τ = A We call this the lifetime of the state ---- technically, it i s the time it takes for 1/ e.368 initial value. a ( t) to reach of its This spontaneous emission formula gives the transition rate for other allowed states. ψ ψ N regardless of any
39 evidently the numer remaining in the excited state decreases exponentially, with a time constant 1 τ = A We call this the lifetime of the state ---- technically, it i s the time it takes for 1/ e.368 initial value. a ( t) to reach of its This spontaneous emission formula gives the transition rate for other allowed states. ψ ψ N regardless of any
40 Typically, an excited atom has many different decay modes ( that is, ψ can decay to a large numer of different lower-energy states, ψ, ψ, ψ L). In that a1 a2 a3 case the transition rates add, and the net lifetime is τ = 1 A + A + A + L While h n x n = ( n δ + nδ ) n, n 1 n, n 1 2mω where ω is the natural frequency of the oscillator.
41 But were talking aout emission, so must e lower than n; for our purpose, then, nh = q ˆ l n, n 1 2m δ ω Evidently transitions occur only to states one step lower on the "ladder", and the frequency of the photon emitted is En E n n + hω n + ω = = h h = ( n n) ω = ω ( 1/ 2) ( 1/ 2) Not surprisingly, the system radiates at the classical oscillator frequency. n hω
42 But the transition rate is A = 2 2 nq ω 6πε mc th and the lifetime of the n stationary state is τ n 6πε mc = 2 2 nq ω Meanwhile, each radiate photon carries an energy so the power radiated is Ahω: P = 2 2 q ω 6πε mc ( nhω) hω,
43 or, since the energy of an oscillator th in the n state is E = ( n + 1) hω, P 2 2 q ω 1 = ( E hω) πε 3 6 mc 2 This is the average power radiated y a quntum oscillator with (initial) energy E. For comparison, lets determine the average power radiated y a classical oscillator with the same energy According to Larmor formula: P = 2 2 q a 6πε c 3
44 For a harmonic oscillator with amplitude x, x( t) = x cos( ω t), and the acceleration is a = x 2 ω cos( ω t). Averaging over a full cycle, then P = q x 12 ω πε c 2 2 But the energy of the oscillator is E = (1 / 2) mω x, so x = 2 E / mω,and herenc 2 2 P = q x 6πε ω mc This is the average power radiated y a classical oscillator with (initial) energy E. E
45 9.3.3 Selective Rules The calculation of spontaneous emission rates has een reduced to a matter of evaluating matrix elements of the form v ψ r ψ a Suppose we are interested in systems like hydrogen, for which the Hamiltonian is spherically symmetrical. In that case we may specify the states with the usual quantum numers n,l, and m, and the matrix elements are nl m r v nlm
46 a) Selection rules involving m and m : Consider first the commutations of L with x, y, and z, which we worked out in Chapter 4: [ L,x] = ihy, [ L,y] = ihx, [ L,z] =. z z z From the third of these it follows that = nl m [ L z,z] nlm = nl m ( Lz z zlz ) nlm = h nl m [( m ) z z( m )) nlm h = ( m m) h nl m z nlm z
47 Conclusion : Either m = m, o r else nl m z nlm =. (1) So unless always zero. m = m, the matrix elements of z are Meanwhile, from the commutator of L with we get z x [ ] = ( z z z nl m L,x nlm nl m L x xl ) nlm = ( m m) h nl m x nlm = ih nl m y nlm
48 Conclusion : ( m m) n l m x = nlm i n l m y nlm Finally, the commutator of L with y yields n l m [ L, y] nlm = n l m ( L y yl ) nlm z z z = = ( m m) h n l m y nlm ih n l m x Conclusion : and ( m m) n l m y nlm z i n l m nlm x nlm 2 ( ) ( ) m m n l m x nlm = i m m n l m y nlm = = n l m x nlm
49 and hence either 2 ( m m) = 1, or else n l m x nlm n l m y nlm = = (2) From Equations (1) and (2), we otain the selection rule for m: No transitions occur unless m = ± 1 or This is an easy result ot understand if you rememer that the photon carries spin 1, and henc its value of m is 1,, or -1; conservation of angular momentum requires that the atom give up whatever the photon takes away.
50 ) Selection rules involving l and l : n l m The commutation relation: v v v [ L,[ L, r ]] = 2 h ( rl + L r ) As efore, we sandwich this commutation etween and nlm to derive the selection rule: 2 n l m [ L,[ L v z,r ]] nlm 2 v 2 2v = 2 h n l m ( rl L r ) nlm 4 v = 2 h [ l( l + 1) + l ( l + 1)] n l m r nlm 2 2 v 2 v 2 = n l m ( L [ L, r ] [ L, r ] L ) nlm 4 2 v = h [ l ( l + 1) + l( l + 1)] n l m r nlm
51 Conclusion Either 2[ l( l + 1) + l ( l + 1)] = [ l ( l + 1) l( l + 1)] v or else n l m r nlm = But and so : 2 [ l ( l + 1) l( l + 1)]=( l + l + 1)( l + 1) 2 2 2[ l( l + 1) + l ( l + 1)]=( l + l + 1) + ( l l) ( l + l + 1) + ( l l) 1= The first factor cannot e zero, so the condition simplifies to l = l ± 1.
52 Thus we otain the selection rule for l: No transitions occur unless l = ± 1 Again, this result is easy to interpret: The photon carries spin 1, so the rules for addition of angular momentum woul d l = l 1. l = l + 1, l = l or
53 Allowed decays for the first four Bohr levels in hydrogen
The Einstein A and B Coefficients
The Einstein A and B Coefficients Austen Groener Department of Physics - Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics III December 10, 010 Abstract In this paper, the Einstein
More informationTime 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 informationB2.III Revision notes: quantum physics
B.III Revision notes: quantum physics Dr D.M.Lucas, TT 0 These notes give a summary of most of the Quantum part of this course, to complement Prof. Ewart s notes on Atomic Structure, and Prof. Hooker s
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
More informationRadiating Dipoles in Quantum Mechanics
Radiating Dipoles in Quantum Mechanics Chapter 14 P. J. Grandinetti Chem. 4300 Oct 27, 2017 P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 1 / 26 P. J. Grandinetti (Chem.
More informationNotes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
More informationAtomic Transitions and Selection Rules
Atomic Transitions and Selection Rules The time-dependent Schrodinger equation for an electron in an atom with a time-independent potential energy is ~ 2 2m r2 (~r, t )+V(~r ) (~r, t )=i~ @ @t We found
More informationINTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM
INTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM I. The interaction of electromagnetic fields with matter. The Lagrangian for the charge q in electromagnetic potentials V
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationApproximation Methods in QM
Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden
More information8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current
Prolem Set 5 Solutions 8.04 Spring 03 March, 03 Prolem. (0 points) The Proaility Current We wish to prove that dp a = J(a, t) J(, t). () dt Since P a (t) is the proaility of finding the particle in the
More information6. Molecular structure and spectroscopy I
6. Molecular structure and spectroscopy I 1 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 light-matter interaction 6.1 molecular spectroscopy introduction 2 Molecular
More informationPreliminary Examination - Day 1 Thursday, August 10, 2017
UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August, 7 This test covers the topics of Quantum Mechanics (Topic ) and Electrodynamics (Topic ). Each topic has 4 A questions
More informationThe energy of the emitted light (photons) is given by the difference in energy between the initial and final states of hydrogen atom.
Lecture 20-21 Page 1 Lectures 20-21 Transitions between hydrogen stationary states The energy of the emitted light (photons) is given by the difference in energy between the initial and final states of
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More information1. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants.
Sample final questions.. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants. 2. A one-dimensional harmonic oscillator, originally in the ground state,
More informationNon-stationary States and Electric Dipole Transitions
Pre-Lab Lecture II Non-stationary States and Electric Dipole Transitions You will recall that the wavefunction for any system is calculated in general from the time-dependent Schrödinger equation ĤΨ(x,t)=i
More informationQuantum Physics III (8.06) Spring 2008 Final Exam Solutions
Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,
More informationQuantum Light-Matter Interactions
Quantum Light-Matter Interactions QIC 895: Theory of Quantum Optics David Layden June 8, 2015 Outline Background Review Jaynes-Cummings Model Vacuum Rabi Oscillations, Collapse & Revival Spontaneous Emission
More information( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1
Physics 624, Quantum II -- Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is
More informationEinstein s Approach to Planck s Law
Supplement -A Einstein s Approach to Planck s Law In 97 Albert Einstein wrote a remarkable paper in which he used classical statistical mechanics and elements of the old Bohr theory to derive the Planck
More informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More information( ) /, so that we can ignore all
Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More information2. Electric Dipole Start from the classical formula for electric dipole radiation. de dt = 2. 3c 3 d 2 (2.1) qr (2.2) charges q
APAS 50. Internal Processes in Gases. Fall 999. Transition Probabilities and Selection Rules. Correspondence between Classical and Quantum Mechanical Transition Rates According to the correspondence principle
More informationPhysics 139B Solutions to Homework Set 4 Fall 2009
Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates
More informationPhysics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.
Physics 505 Homework No 8 s S8- Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will
More informationThe interaction of electromagnetic radiation with one-electron atoms
The interaction of electromagnetic radiation with one-electron atoms January 21, 22 1 Introduction We examine the interactions of radiation with a hydrogen-like atom as a simple example that displays many
More informationLight - Atom Interaction
Light - Atom Interaction PHYS261 fall 2006 Go to list of topics - Overview of the topics - Time dependent QM- two-well problem - Time-Dependent Schrödinger Equation - Perturbation theory for TDSE - Dirac
More informationPHY4604 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 informationCh 125a Problem Set 1
Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract
More informationMolecular spectroscopy
Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering
More informationLight - Atom Interaction
1 Light - Atom Interaction PHYS261 and PHYS381 - revised december 2005 Contents 1 Introduction 3 2 Time dependent Q.M. illustrated on the two-well problem 6 3 Fermi Golden Rule for quantal transitions
More informationQuantum 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 informationa = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam
Physics 624, Quantum II -- Final Exam Please show all your work on the separate sheets provided (and be sure to include your name). You are graded on your work on those pages, with partial credit where
More informationIntroduction 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 informationLINEAR RESPONSE THEORY
MIT Department of Chemistry 5.74, Spring 5: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff p. 8 LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior
More informationScattering of Electromagnetic Radiation. References:
Scattering of Electromagnetic Radiation References: Plasma Diagnostics: Chapter by Kunze Methods of experimental physics, 9a, chapter by Alan Desilva and George Goldenbaum, Edited by Loveberg and Griem.
More informationSo far, we considered quantum static, as all our potentials did not depend on time. Therefore, our time dependence was trivial and always the same:
Lecture 20 Page 1 Lecture #20 L20.P1 Time-dependent perturbation theory So far, we considered quantum static, as all our potentials did not depend on time. Therefore, our time dependence was trivial and
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More information2 Quantization of the Electromagnetic Field
2 Quantization of the Electromagnetic Field 2.1 Basics Starting point of the quantization of the electromagnetic field are Maxwell s equations in the vacuum (source free): where B = µ 0 H, D = ε 0 E, µ
More information14 Time-dependent perturbation theory
TFY4250/FY2045 Lecture notes 14 - Time-dependent perturbation theory 1 Lecture notes 14 14 Time-dependent perturbation theory (Sections 11.1 2 in Hemmer, 9.1 3 in B&J, 9.1 in Griffiths) 14.1 Introduction
More informationQuantization of the E-M field
April 6, 20 Lecture XXVI Quantization of the E-M field 2.0. Electric quadrupole transition If E transitions are forbidden by selection rules, then we consider the next term in the expansion of the spatial
More informationThe interaction of light and matter
Outline The interaction of light and matter Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, 014 1 / 3 Elementary processes Elementary processes 1 Elementary processes Einstein relations
More informationQuantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths
Quantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths 4.1 The Natural Line Shape 4.2 Collisional Broadening 4.3 Doppler Broadening 4.4 Einstein Treatment of Stimulated Processes Width
More informationReading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom.
Chemistry 356 017: Problem set No. 6; Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom. The H atom involves spherical coordinates and angular momentum, which leads to the shapes
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationPAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)
Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE
More informationψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.
1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product
More informationPhysics 221B Spring 2012 Notes 42 Scattering of Radiation by Matter
Physics 221B Spring 2012 Notes 42 Scattering of Radiation by Matter 1. Introduction In the previous set of Notes we treated the emission and absorption of radiation by matter. In these Notes we turn to
More informationCollection 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 informationPerturbation Theory. Andreas Wacker Mathematical Physics Lund University
Perturbation Theory Andreas Wacker Mathematical Physics Lund University General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationQUALIFYING EXAMINATION, Part 1. 1:00 pm 5:00 pm, Thursday August 31, 2017
QUALIFYING EXAMINATION, Part :00 pm 5:00 pm, Thursday August 3, 207 Attempt all parts of all four problems. Please begin your answer to each problem on a separate sheet, write your 3 digit code and the
More informationEE201/MSE207 Lecture 10 Angular momentum
EE20/MSE207 Lecture 0 Angular momentum (A lot of mathematics; I will try to explain as simple as possile) Classical mechanics r L = r p (angular momentum) Quantum mechanics The same, ut p x = iħ x p =
More information4. Spontaneous Emission october 2016
4. Spontaneous Emission october 016 References: Grynberg, Aspect, Fabre, "Introduction aux lasers et l'optique quantique". Lectures by S. Haroche dans "Fundamental systems in quantum optics", cours des
More informationMath Questions for the 2011 PhD Qualifier Exam 1. Evaluate the following definite integral 3" 4 where! ( x) is the Dirac! - function. # " 4 [ ( )] dx x 2! cos x 2. Consider the differential equation dx
More information8 Quantized Interaction of Light and Matter
8 Quantized Interaction of Light and Matter 8.1 Dressed States Before we start with a fully quantized description of matter and light we would like to discuss the evolution of a two-level atom interacting
More information16.1. 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 informationPhysics 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 informationNotes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14
Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Because the wavelength of light (400-700nm) is much greater than the diameter of an atom (0.07-0.35
More informationThe Simple Harmonic Oscillator
The Simple Harmonic Oscillator Michael Fowler, University of Virginia Einstein s Solution of the Specific Heat Puzzle The simple harmonic oscillator, a nonrelativistic particle in a potential ½C, is a
More information2.1 Green Functions in Quantum Mechanics
Chapter 2 Green Functions and Observables 2.1 Green Functions in Quantum Mechanics We will be interested in studying the properties of the ground state of a quantum mechanical many particle system. We
More informationBohr Model. In addition to the atomic line spectra of single electron atoms, there were other successes of the Bohr Model
Bohr Model In addition to the atomic line spectra of single electron atoms, there were other successes of the Bohr Model X-ray spectra Frank-Hertz experiment X-ray Spectra Recall the x-ray spectra shown
More informationQuantum mechanics (QM) deals with systems on atomic scale level, whose behaviours cannot be described by classical mechanics.
A 10-MINUTE RATHER QUICK INTRODUCTION TO QUANTUM MECHANICS 1. What is quantum mechanics (as opposed to classical mechanics)? Quantum mechanics (QM) deals with systems on atomic scale level, whose behaviours
More informationEigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator.
PHYS208 spring 2008 Eigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator. 07.02.2008 Adapted from the text Light - Atom Interaction PHYS261 autumn 2007 Go to list of topics
More informationCreation and Destruction Operators and Coherent States
Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function state harmonic oscillator wave function, We first rewrite the ground < x 0 >= ( π h )1/4 exp( x2 a 2 h )
More informationECE 240a - Notes on Spontaneous Emission within a Cavity
ECE 0a - Notes on Spontaneous Emission within a Cavity Introduction Many treatments of lasers treat the rate of spontaneous emission as specified by the time constant τ sp as a constant that is independent
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 9 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Andrei Tokmakoff,
More informationQuiz 6: Modern Physics Solution
Quiz 6: Modern Physics Solution Name: Attempt all questions. Some universal constants: Roll no: h = Planck s constant = 6.63 10 34 Js = Reduced Planck s constant = 1.06 10 34 Js 1eV = 1.6 10 19 J d 2 TDSE
More informationWe now turn to our first quantum mechanical problems that represent real, as
84 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like
More information8.05 Quantum Physics II, Fall 2011 FINAL EXAM Thursday December 22, 9:00 am -12:00 You have 3 hours.
8.05 Quantum Physics II, Fall 0 FINAL EXAM Thursday December, 9:00 am -:00 You have 3 hours. Answer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white books. There
More informationPhysics 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 informationin terms of the classical frequency, ω = , puts the classical Hamiltonian in the form H = p2 2m + mω2 x 2
One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part because its properties are directly applicable to field theory. The treatment in Dirac notation is particularly
More informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
More informationQuantum Fysica B. Olaf Scholten Kernfysisch Versneller Instituut NL-9747 AA Groningen
Quantum Fysica B Olaf Scholten Kernfysisch Versneller Instituut NL-9747 AA Groningen Tentamen, vrijdag 3 november 5 opgaven (totaal van 9 punten. Iedere uitwerking op een apart vel papier met naam en studie
More informationChapter 1 Early Quantum Phenomena
Chapter Early Quantum Phenomena... 8 Early Quantum Phenomena... 8 Photo- electric effect... Emission Spectrum of Hydrogen... 3 Bohr s Model of the atom... 4 De Broglie Waves... 7 Double slit experiment...
More informationToday: general condition for threshold operation physics of atomic, vibrational, rotational gain media intro to the Lorentz model
Today: general condition for threshold operation physics of atomic, vibrational, rotational gain media intro to the Lorentz model Laser operation Simplified energy conversion processes in a laser medium:
More informationAdvanced Optical Communications Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay
Advanced Optical Communications Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture No. # 15 Laser - I In the last lecture, we discussed various
More informationSection 5 Time Dependent Processes
Section 5 Time Dependent Processes Chapter 14 The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic,
More informationAtkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 7: Quantum Theory: Introduction and Principles
Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas Chapter 7: Quantum Theory: Introduction and Principles classical mechanics, the laws of motion introduced in the seventeenth century
More informationPhysics 342 Lecture 17. Midterm I Recap. Lecture 17. Physics 342 Quantum Mechanics I
Physics 342 Lecture 17 Midterm I Recap Lecture 17 Physics 342 Quantum Mechanics I Monday, March 1th, 28 17.1 Introduction In the context of the first midterm, there are a few points I d like to make about
More informationPHYS 3313 Section 001 Lecture # 22
PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for
More informationTime Independent Perturbation Theory Contd.
Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n
More informationAccelerator Physics NMI and Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 16
Accelerator Physics NMI and Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Graduate Accelerator Physics Fall 17 Oscillation Frequency nq I n i Z c E Re Z 1 mode has
More information129 Lecture Notes Relativistic Quantum Mechanics
19 Lecture Notes Relativistic Quantum Mechanics 1 Need for Relativistic Quantum Mechanics The interaction of matter and radiation field based on the Hamitonian H = p e c A m Ze r + d x 1 8π E + B. 1 Coulomb
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
More informationEnergy during a burst of deceleration
Problem 1. Energy during a burst of deceleration A particle of charge e moves at constant velocity, βc, for t < 0. During the short time interval, 0 < t < t its velocity remains in the same direction but
More informationTotal 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 information3. Quantum Mechanics in 3D
3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary
More informationLaser Cooling of Gallium. Lauren Rutherford
Laser Cooling of Gallium Lauren Rutherford Laser Cooling Cooling mechanism depends on conservation of momentum during absorption and emission of radiation Incoming photons Net momentum transfer to atom
More informationPreliminary Examination - Day 2 Friday, May 11, 2018
UNL - Department of Physics and Astronomy Preliminary Examination - Day Friday, May, 8 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Quantum Mechanics (Topic ). Each
More informationCONTENTS. vii. CHAPTER 2 Operators 15
CHAPTER 1 Why Quantum Mechanics? 1 1.1 Newtonian Mechanics and Classical Electromagnetism 1 (a) Newtonian Mechanics 1 (b) Electromagnetism 2 1.2 Black Body Radiation 3 1.3 The Heat Capacity of Solids and
More information11 Perturbation Theory
S.K. Saikin Oct. 8, 009 11 Perturbation Theory Content: Variational Principle. Time-Dependent Perturbation Theory. 11.1 Variational Principle Lecture 11 If we need to compute the ground state energy of
More informationProblem 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 informationMATH325 - QUANTUM MECHANICS - SOLUTION SHEET 11
MATH35 - QUANTUM MECHANICS - SOLUTION SHEET. The Hamiltonian for a particle of mass m moving in three dimensions under the influence of a three-dimensional harmonic oscillator potential is Ĥ = h m + mω
More information1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q.
1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q. (a) Compute the electric part of the Maxwell stress tensor T ij (r) = 1 {E i E j 12 } 4π E2 δ ij both inside
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationClassical electric dipole radiation
B Classical electric dipole radiation In Chapter a classical model was used to describe the scattering of X-rays by electrons. The equation relating the strength of the radiated to incident X-ray electric
More information