# PH 451/551 Quantum Mechanics Capstone Winter 201x

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1 These are the questions from the W7 exam presented as practice problems. The equation sheet is PH 45/55 Quantum Mechanics Capstone Winter x TOTAL POINTS: xx Weniger 6, time There are xx questions, for a total of xx points. They will surely present different levels of difficulty, so work quickly on the ones you find easy, leaving time for the ones you find harder. On average, budget about point per minute. Equations and integrals that may be useful: ψ Â φ = φ Â ψ * n n = ΔQ Q Q n ( ) π E n = n ml E = n n α mc E n = ( n + )ω (x) = L sin nπx L α = e 4πε c a = 4πε m e e ( ) = x a = V x p x + i x a = p x i x a n = n n a n = n + n + E n = ( n + )ω ϕ ( x) = π (x) = n/ n π 4 e x ϕ x /4 H n ( ) = x E () n = n ( ) H n ( ) E () n = π e x m n 4 n ( ) H m ( ) E n () E m () xe n ( ) ( ) = c nm m ( ) ( c ) nm = m ( ) H n ( ) H ' = d i Ε m n E () () n E m µ = g q S µ B = e µ N = e m m e m p d = qr S x S y i i S z S 4 x

2 These are the questions from the W7 exam presented as practice problems. The equation sheet is S x S y i i i i S z S J = j( j +) J z = m j J x, J y = ij z J ± = j j + ( ) ( ) m j m j ± JM = j j m m j j m m JM m m J + J = J + J J + J H hf = A SiI J ± = J x ± ij y ± H rel = p4 8m c H = e SO 8πε m c r LiS H Z = µ i B = µ BB ( g L z + g e S z ) ( E ) fs = α 4 mc n j + c f ( t) = i t ( ) l( l +) 4 ( )( l +) 4n E SO = 4 α j j + 4 mc n l l + E f E i t f H ( t ) i e i d t R i f = π H ψ = i t ψ H = E n e a x dx = a π xe x dx = x n e ax dx = n a n+ ψ t= = n V fi c n x e x dx = π 4 sinθ = sinθ cosθ cosθ = sin θ tanθ = tanθ tan θ sin mxsin nx dx = sin( m n) x sin( m + n) x, ( m ( m n) ( m + n) n ) cos mx cosnx dx = sin( m n)x + sin( m + n)x, ( m n ) m n m + n sin mx cosnx dx ( ) ( ) x ( m n) = cos m n ( ) ( ) x ( m + n) cos m + n, ( m n ) δ ( ω fi ω )

3 These are the questions from the W7 exam presented as practice problems. The equation sheet is s= F j = j = j m m s I = m I M F - m m j = j j = j j = m m j = m - m m m

4 These are the questions from the W7 exam presented as practice problems. The equation sheet is 4

5 These are the questions from the W7 exam presented as practice problems. The equation sheet is. [ points] Perturbed HO A particle is bound in the harmonic oscillator potential V x ( ) = x. A static perturbation H = γ x is applied to the system. (a) Calculate the first-order corrections to (all) the energy eigenvalues using the perturbation formalism. (b) What is the exact energy? Explain why your answer is consistent with (a) above.. [ points] Degenerate perturbation theory A particle in a two-dimensional harmonic oscillator potential U ( x, y) = ( x + y ) has energy eigenstates that are characterized by TWO quantum numbers n x and n y, each of which can have values,,,,. The energy eigenstates are products of the -d energy eigenstates: n x,n y x ( x)y ( y) with corresponding energy eigenvalues E nx,n y = ( n x + )ω + ( n y + )ω = ( n x + n y +)ω (a) (5 points) What is the energy of the first excited state (the next state above the lowestenergy state)? Why is this energy level degenerate, and what is the degeneracy of the level? (b) (5 points) Calculate the new energy eigenvalues and eigenstates (to first order) of these degenerate states that comprise the first excited state when a small perturbation to the Hamiltonian of the form H '( x) = U ( ) is applied where δ ( x) is the Dirac delta δ x function. Proceed as follows: Construct the matrix that represents the perturbation H' in the degenerate subspace and use it to determine the first-order changes, if any, to the energy eigenvalues and the eigenstates.. [ points] Fine structure In the simplest treatment of the hydrogen atom, the electron and proton experience only the Coulomb interaction, and the energy eigenstates are designated n l m l s m s. (a) What are the allowed values of these quantum numbers? (b) If the system is in the p state, what are the particular values of the quantum numbers and what is the (unperturbed) energy of the p state? What is its degeneracy? (c) In class we used degenerate perturbation theory to show that adding interactions to the Hamiltonian introduced fine structure to the H-atom spectrum and that the finestructure corrections are E ) fs ( = α 4 mc n j + 4n. Use this expression to show how the unperturbed p levels are changed: draw a diagram that shows the new level(s), correctly labeled, and find the size of the correction(s) in ev. (d) What wavelength resolution is necessary to resolve the fine structure in the s < > p transition? Give a fractional value rather than an absolute wavelength. 5

6 These are the questions from the W7 exam presented as practice problems. The equation sheet is 4. [ points] Two spin systems Two particles A and B form the quantum system of interest. Only the spin angular momentum is relevant in this problem. A is spin-/ particle and B is a spin- particle. (a) Write down all possible kets that describe the state of system in the (i) coupled, and (ii) uncoupled bases. (State what the numbers in the kets represent in both cases). (b) Use the Clebsch-Gordan coefficients to write the coupled states as linear combinations of the uncoupled states. (c) If the system is in the state where the total angular momentum is given by j = /, and the z-component of the total angular momentum is m j = /, what is (and explain why) the probability that a measurement of the z-component of the angular momentum of B yields: (i), (ii), (iii)? 5. [ points] Identical particles Two indistinguishable, non-interacting, uncharged spin-/ fermions are confined to a region of length L in one-dimension. In your answers, explain your reasoning. (a) Find the energy eigenvalue(s) of the ground state. (b) Find the energy eigenstate(s) of the ground state. (c) Specify and discuss the degeneracy of the ground state. (d) Find the energy eigenvalue(s) of the first excited state. (e) Find the energy eigenstate(s) of the first excited state. (f) Specify and discuss the degeneracy of the first excited state. END OF EXAM 6

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