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

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1 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, * x x dx N exp x i x exp x i x dx N x dx N N /4 (b) [9] What is X and P for this state? exp exp exp, * X x x x dx x ix x x ix dx N x x dx d dx exp exp * P x i x dxi x ix x ix dx in xiexpx dx i i i i.. [3 points] In a three-dimensional space with orthonormal basis x, y,, the operator L has the following properties: Lx iy, Ly ix, L. (a) [] Write L as a 33matrix. Is it Hermitian? d dx This is obviously Hermitian. xlx xly xl i L ylx yly yl i Lx Ly L (b) [] Find the eigenvalues and orthonormal eigenvectors of L. We first note that L is block diagonal, as marked by the dashed lines. This reduces the problem to two problems, one of which has a trivial eigenvector in the third component with

2 eigenvalue, and two of which are non-trivial. To find the non-trivial ones, we first work on the characteristic equation for the smaller submatrix, which is i ii i det, To find these eigenvectors, we write down the eigenvector equation i i or i i i and i or i and i i or i In either case, to get the state properly normalied, we must take that we must have, so we pick and we find our two eigenstates, which implies,. i i Reverting back to the full three-dimensional system, and including our third eigenvector, the eigenvectors are then i, i,. (c) [7] A system is in the state x when L is measured. What are the possible outcomes of that measurement, and the corresponding probabilities? The probability of an outcome a is just a summed up over all the possible eigenstates with the same eigenvalue. Since there is only one state with each eigenvalue, there is no sum. It is clear that vanishes, so this is not a possible outcome, whereas, so P. (d) [6] For each of the outcomes in part (c), what will the state vector look like afterwards? After a measurement, the state will naturally be in a normalied eigenstate of the operator with the corresponding eigenvalue. This means that after measuring the result, we will, up to an irrelevant phase factor, be in the state.

3 3. [ points] Consider the harmonic oscillator with mass m and angular frequency. 3 4 For the quantum state i, find X and P. X X i aa i m i i i i i, m m 3 3 P P mi i a a i mi i i i m i i i m 4 m. 4. [ points] A particle of mass m lies in the infinite square well with allowed region x a. At t =, the wave function takes the form x, t a in the allowed region and it vanishes elsewhere (a) [8] Write this state in the form c n n n, where n is the n th eigenstate of the Hamliltoion. Work out a formula for the c n s. The cn s are given by * a nx a nx cn n n xt, xdx sin dx cos a a a a n a n n n odd cos n n n n even We therefore have (b) [] Check that in the new basis, n n n odd is properly normalied. For this purpose, you can just keep a couple non-ero terms and make sure the sum is coming out about right. We simply need to check if 8 cn. n n odd n Adding the first three non-vanishing terms, the sum is.933, which is tending towards one. Getting the exact sum is harder. a

4 t as a function of time in terms of the eigenstate basis, and x, t (c) [7] Write You are not expected to do the sums. Each term just gets multiplied by the phase factor expie t n., so our final answer is The wave function is n t exp i t n n odd n ma n nx 4 xt, expi tsin. n odd na ma a. [ points] The angular momentum operator L has the commutation properties L, X i Y and L, Y i X. Define two new operators A X Y and B XY. (a) [] Prove the following true statements: L, A i B or L, B i A We simply have: L, A L, X Y L, X L, Y,,,, X L X L X X Y L Y L Y Y ixy iyx iyx ixy ixy ib, L, B L, XY X L, Y L, X Y i XX i YY i X Y ia. Looks like they are both true. (b) [] Based on the commutation relations found in (a), write two uncertainty relations. The uncertainty relations are: L A i L, A iib B L B i L, B iia A

5 6. [ points] A particle of mass in two dimensions lies in the potential V A X Y. (a) [] What continuous symmetry and corresponding generator commutes with this Hamiltonian? What are the corresponding possible eigenvalues of this generator? Are there any restrictions on this eigenvalue? It is clear that no translation commutes with this Hamiltonian, but we know that the distance from the origin is unchanged under rotation (around the fictitious -axis, since we are in two dimensions), so rotation commutes with this Hamiltonian, and hence so does its generator, L. We normally write states as eigenstate of L, so L m. When written this way, the restriction on m is that m must be an integer. (b) [] Write the wave function as, R one of these explicitly. in polar coordinates. Write The L operator, written in polar coordinates, is L i i R m R im, im e,, so we have (c) [] Deduce an ordinary differential equation for the other one. The D Laplacian in polar coordinates is given below. Do not attempt to solve this equation. Schrödinger s equation in D, when expanded in polar coordinates, is E H V X, Y Substituting our explicit form for our wave function in, we have ER e Re R e Re V Re m im im R R R e ARe, d d m ER R R A R. d d im im im im im We can replace the partial derivatives by regular derivatives because R is a function of only.