Physics 742 Graduate Quantum Mechanics 2 Midterm Exam, Spring [15 points] A spinless particle in three dimensions has potential 2 2 2

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1 Physics 74 Graduate Quantum Mechanics Midterm Exam, Spring 4 [5 points] A spinless particle in three dimensions has potential V A X Y Z BXYZ Consider the unitary rotation operator R which cyclically permutes the position operators x, y, and z, so that in the position basis, we would h R xyz,, yzx,,, or equivalently: R XR Y R YR Z R ZR X,,, and similar equations apply to the momentum as well (a) [6] Convince yourself (and me) that R commutes with the Hamiltonian The momentum term is invariant under all rotations The distance squared is also invariant, as is easily demonstrated, for example R X Y Z R RXR RYR RZR RXRRXR RYRRYR RZRRZR We therefore h YY ZZ XX X Y Z RVR AR X Y Z RBR XYZR AR X Y Z RR X Y Z R BR XRR YRR ZR A X Y Z X Y Z BYZX V So R VR V, or multiplying on the left by R, we h VR RV, and hence the whole Hamiltonian commutes with R (b) [6] Show that R 3 = Find all possible eigenvalues of R It is pretty easy to see that 3 R xyz R yzx Rzxy xyz,,,,,,,,, 3 which, since x, yz, forms a complete basis set, shows that R If we h any 3 3 eigenstate such that R, then it is easy to see that R, 3 which implies that 3, and we can then solve this equation to find the three possible eigenvalues namely This can be rewritten as or i 3

2 (c) [3] Certain of the eigenvalues you found in part (b) will result in twofold degeneracies of the Hamiltonian Which eigenvalues is this true for? Let be an eigenstate of the Hamiltonian, so that H E If we think in a coordinate basis, then H is real, as is E, and hence H E, where is the state vector whose w function is the complex conjugate of We also know that R, but thinking in a coordinate basis again, R is purely real, so that if we take the complex conjugate of this relation, we h R If (which means ), then this doesn t tell us anything; indeed, it is likely that and are the same thing, but if, then this means that and are 3 orthogonal, and hence must be different states It follows that for i, we will get pairs of states, and, with identical energies [ points] The first two eigenstates of the Harmonic oscillator h w functions of the form /4 /4 Ax /4 3/4 Ax x x A e x x A xe and (a) [3] For two non-identical non-interacting particles, write the normalized w function x, x if the first particle is in state n = and the second in n = Normalized states of two particles just look like tensor products of the individual states, and hence are of the form,, or in this case,,, and the resulting w functions are just direct products, so Ax x x, x x, x, x x Ax e (b) [5] What is the probability that both particles h xi simultaneously? Taking advantage of the given integrals, this is given by,, Ax Ax / 3/ A A 4 A 4 P x x x x dxdx A e dx x e dx

3 (c) [5] Repeat part (a), but this time assume the two particles are (identical spin- ½) electrons in the spin state I want x, x; properly normalized This time we need to make the w function anti-symmetric, because electrons are fermions This is done simply by making,,,;, ; Writing out the properly normalized w function, we then h x, x ; x, x, x, x, x x x x Ax x Ax x Ax x Ax e Ax e A x x e (d) [7] What is the probability that both particles h xi simultaneously? We simply perform the same computation as before, except the w function is more complicated, so we get more terms,, Ax Ax A Ax Ax Ax Ax Ax Ax e dx x e dx x e dx e dx xe dx x e dx P x x x x dxdx A x x e dxdx A A A A A A / 3/ 3/ / 4 4 A [ points] An electron lies in a region with magnetic and electric fields axˆ yˆ, U u x y z A r y x r where a and u are positive constants (a) [5] What are the electric and magnetic fields from these potentials? We calculate these using the given formulas: A Ay z Ax Az Ay A x BAxˆ yˆ zˆ zˆa a azˆ, y z z x x y A U U U EU xˆ yˆ zˆ uxxˆ uyyˆ 4uzzˆ t x y z

4 (b) [5] Write the Hamiltonian for an electron in these fields We simply substitute directly into the formula for the Hamiltonian: ge H e, t eu, t, t m P A R R B R S m gea P eay P eax P eu X Y Z S m m Px Py Pz eaypx eaxpy e a X e a Y m gea eu X Y Z Sz m x y z z (c) [] Show that the Hamiltonian you found in part (b) into something of the form H P cl m xx yy zz ds, m and determine the vectors c and d and the three frequencies i Determine the range of value of u for which all three frequencies i are positive It is a moment s work to rewrite the expression above at ea e a gea H P Lz eu X Y euz Sz m m m m We therefore h ea gea e a c zˆ, d z ˆ, mx my eu, mz eu m m m These last two equations can be solved, if we want to: e a eu 4eu x y, z m m m ea These frequencies will all be positive provided u m (d) [3] Find a spin and an angular momentum operator that commute with H You don t h to prove they commute The spin operator Sz and Lz both commute with H This is obvious for Sz For Lz, we simply note that the quantities X Y and Z are both unchanged by rotations around the z-axis

5 4 [ points] Consider the ground state of the Harmonic oscillator (a) [5] Find the expectation value of the position operator X, the position squared X, and the uncertainty X for a single particle in the ground state of the Harmonic oscillator If we were to make a measurement of a single particle, can we predict what value x will be obtained? If so, what value is it, or if not, predict what value it would rage to if many measurements were made We can easily see that X a a m m m We therefore h X X, m X X X X, m m m X X X m Because the uncertainty is non-zero, this implies that this is not an eigenstate of X However, over a series of measurements, the value x should rage to X (b) [5] Now consider a large collection of N particles, all in the ground state of the Harmonic oscillator, Define the rage position operator X N N i X Find the expectation value of X, X, and the uncertainty X for this quantum state In the limit N, if we measured X, can we predict what value x will be obtained, and if so, what value would we obtain? We similarly first find We therefore h X a a N i i N m i N m i

6 X X, N m X X X X N m NN N N m Nm X X X Nm In the limit, N, the uncertainty drops to zero, which implies we are in an eigenstate of X, and the value we will get will be x X 5 [ points] A particle of mass m in one dimension lies in potential V x A x, where A is a positive constant The bound states will h negative energy, En Estimate the energy of the n th bound state by using the WKB approximation The first step is to locate the classical turning points; those points where V xa x E This will be the points where x AE A, or x A We therefore h A A A A n me Vx dx m A x dx m dx A A x x A A x m sin Ax x A x A A A A A m sin A m sin A A m A m, A m, n ma En n n

7 6 [5 points] A particle of mass m scatters from a weak potential V Ve r Particles are incident on it with momentum k Find the differential crosssection d d as a function of the angle of deflection, and the total cross section We first need to calculate the Fourier transform of V Because the potential is spherically symmetric, it doesn t matter which direction K is in, so we might as well put it along the z-axis So we h Kr Kr K r r r cos V e V d V e e d V d d e e r dr i 3 i ir 3 ikrcos r cos 3 V V ik d ik ik cos V V ik ik V ik cos cos cos 4 ik ik ik ik K ik K 8V K We then substitute this into the formula for the differential cross-section 3 ikr 8V 6m V d rv 4 re 4 4 d m m d 4 4 K k k cos We then simply integrate this over angles to yield 4 d 6mV 4 d d 4 k k cos dcos d 3 cos 3mV 4 k k cos, 3 k cos 6mV k 4k 3

8 Helpful Formulas: EM Fields BA E AU t Gauge Transformation U U t A A ie e Kinematic Momentum π PeA EM Hamiltonian ge H π eu m BS m Rotation Operators: R r r R R R RR R b a WKB Quantization mev xdxn First Born Approximation d 3 i d V 4 e d 4 r r K k cos Symmetrization P N! P PP N! Kr P Harmonic Oscillator X P i aa m m a a an nn a n n n Angular Momentum Lx YPz ZPy Ly ZPx xpz L XP YP z y x Helpful Integrals: Gaussian: ax / ax ax 3/ e dx a xe dx a x e dx 4 a,,, 3 ax 4 ax 3 5/ 5 ax 3 x e dx a x e dx 8 a x e dxa,, Exponential: ax n ax n e dxa, x e dxn! a ax b dx ax b C, n a n n Power Law: Radical: 3/ a bx ax b dx ax b C, a x b dx sin ax bx C 3a b a dx dx a bx sin ax b a a x b b b a b ax b C ax bx C