Physics 606, Quantum Mechanics, Final Exam NAME ( ) ( ) + V ( x). ( ) and p( t) be the corresponding operators in ( ) and x( t) : ( ) / dt =...

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1 Physics 606, Quantum Mechanics, Final Exam NAME Please show all your work. (You are graded on your work, with partial credit where it is deserved.) All problems are, of course, nonrelativistic. 1. Consider a particle in one dimension with position operator x and momentum operator p. In doing this problem, start with the fundamental commutation relation involving x and p -- i. e., the fundamental equation for x, p [ ]. (Do not start with any less fundamental equation.) You may use the identity where A, B, C are any operators. [ A, BC] = [ A, B]C + B[ A,C] (a) (5) Show that!" x, p n # $ = constant % p n&1,!" p, x n # $ = different constant % x n&1 while determining these n -dependent constants at the same time. Use a proof by induction, showing (i) that each equation is true for n = 1 and (ii) if it is true for n! 1 then it is true for n. (b) (5) Let f ( p) and g( x) be any functions that can be expressed as power series. Show that!" x, f ( p) #$ = constant % d f p dp while determining the constants at the same time.,!" p,g( x) #$ dg x = different constant % dx (c) (5) The Hamiltonian for this particle has the form H = T p The above operators are in the Schödinger picture. Now let x t the Heisenberg picture. Obtain the equations of motion for p t dp t + V ( x). and p( t) be the corresponding operators in and x( t) : / dt =... / dt =... dx t

2 (d) (5) In the special case T ( p) = p 2 / 2m, obtain a simple equation of the form d 2 x / dt 2 = function of x where!!! denotes the expectation value of an operator, at the same time determining this function of x. (e) (5) Let! be a distance corresponding to a translation in space. Use the result of Part (b) to show that e ip!/! x e "ip!/! = x + constant while also determining the constant. (Note that the operator here is T = e ip!/!.) (f) (5) Then calculate the value of X ' for the Hilbert space vector X ' = X e ip!/! and for the wavefunction! ( X ') = X '! = X e ip"/!! that is obtained when the state! is transformed according to! "! ' = e ip#/!!. (I.e., relate X ' to the original value X in the wavefunction! X before the state is transformed.) Write down the infinitesimal version of this transformation. What is its generator?

3 2. (a) (9) Explicitly construct the three 3! 3 matrices that represent the angular momentum operators L x, L y, L z in the space of the! = 1 functions: where i = x, y, z. Recall that m,m' = 1m L i 1m' L i L ± = L x ± il y and L ±! m = "!! + 1! m( m ± 1)! m ± 1. (b) (8) Using your matrix representation, calculate the commutator!l " y, L z # $. Is your result correct? (c) (8) For a spherically symmetric atomic potential and a state with! > 0, determine the dependence of the radial wavefunction R r on the radial coordinate r near the nucleus. I.e., calculate R( r) as r! 0 up to a normalization constant. (Obtain the function and do not just say 0 of course!) The radial Schrödinger equation is and the potential V r " $! 1 # r! 2 2m d 2 dr 2 r +!2 2m "(" + 1) + V ( r) r 2 % ' & R r = ER( r) can be assumed to reduce to a Coulomb potential as r! 0.

4 3. The Stark effect for the n = 2 states of hydrogen must be treated with degenerate perturbation theory. An electric field is applied along the z axis and produces a shift in the potential energy. As we showed in class, the only relevant nonvanishing matrix element is 2s 0 z 2 p 0 = 2 p 0 z 2s 0 =!3a 0 where a 0 is the Bohr radius. (a) (10) Calculate the magnitude of the shift in energy due to the electric field, and give your answer in terms of the electric field E, a 0, and the fundamental charge e. (Always recall that you are graded on your work.) (b) (10) Calculate the new states after this breaking of the degeneracy by the electric field. Express each of the 2 new states as a linear combination of 2s 0 and 2 p 0, and clearly indicate which state goes with each energy shift.

5 4. (a) (5) Write down the Schrödinger equation for a particle moving in 1 dimension in an arbitrary potential V x. Then rewrite it in terms of #! 2m k x $ % ( )! 2 E " V x & ' ( 1/2. (b) (5) Write the wavefunction in the form! x = e iu x and then rewrite the second equation in Part (a) as a nonlinear equation in u x. (c) (5) Initially neglecting d 2 u / dx 2, solve for du / dx. Then obtain u '' x in terms of k '( x)! dk( x) / dx.! d 2 u / dx 2 (d) (5) Now put this solution for u'' x solution for du / dx. (I.e., now, instead of simply neglecting u'' x by the approximate value from Part (c) in an iterative approach.) back into the nonlinear equation of Part (b), and obtain an improved in the equation of Part (b), we replace it (e) (5) Integrate the result of Part (d) for du / dx, using the approximation k ' x square root. You should finally obtain the WKB wavefunctions! ( x) = " e±i k x 1 k x with k( x) imaginary in a classically forbidden region. dx! k( x) 2 in treating the Happy Holidays!