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(PHYS234)[2008](s)final~ph_kpx^_10546.pdf
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PHYS234 Elementary Quantum Mechanics I Sample Final Examination Questions
Formulae:
NOTE: not every formula listed here is needed in answering this exam paper
1D time-independent Schr.dinger equation (1) Time-dependence of stationary state (2) First quantization rule: , (3) Eigenstates of momentum operator (plane waves): (4) Eigenstates of position operator : (5) Commutation relation (6) Time evolution of expectation value 222()()()2dVxxExmdx............
/(,)()iEtxtex.....
.xx.
.dpidx...
.p
/12ipxpe....
.x
'(')xxx...
[,][,][,]ABCBACABC..

(7) 3D time-independent Schr.dinger equation with a central potential: ..,dAiAHAdtt.....

(8) Laplacian operator in Cartesian and spherical coordinates: 22()2VrEm........

(9) Solution of the hydrogen atom: 222222222222.1Lrxyzrrrr.......................

(10) First few normalized functions are: (11) where is the Bohr radius. Energy levels of a hydrogen atom: ()()(,)ssmsnlmmnllmnlmmRrY.......r
/103/2/2203/2/2213/2211122124rararaRearReaarReaa.............
1/2001/2011/211143cos43sin8iYYYe..............................
2204/ame....

(12) Normalization of radial and angular parts and (13) Orbital angular momentum operators in spherical coordinates: 2222113.6eV2nEmann.....
220()1Rrrdr...
2200sin1mlYdd........

(14) Commutation relations of the angular momentum operators: (15) Eigenstates and eigenvalues of and : (16) Coupling of two spin s = 1/2 particles, : 2222211.sinsinsin.zLLi..................................
2.........[,], [,], [,]..[,]0 where ,,xyzyzxzxyLLiLLLiLLLiLLLxyz..........
2.L
.zL
22..(1), .mmmmllzllLYllYLYmY.....
sSm

(17) 121200() singlet1111 triplet10()........................

Question 1 For a general quantum mechanical problem with Hamiltonian H, we usually denote its normalized stationary states as , where a, b, c, are good quantum numbers corresponding to eigenvalues of operators A, B, C, respectively. ...abc

a) State the properties of the operators A, B, C, . No mathematical expressions are necessary.

b) Without knowing the mathematical form of the wavefunctions , argue that they must be orthogonal, i.e., . aabbccabcabc.............



An electron of a hydrogen atom is in the superposition state , (18) where a is the Bohr radius, A is a normalization constant, and are spherical harmonics. ..//21101113/2274raraerAeiYYYaa....................
mlY

c) First express as a linear combination of hydrogen atom solution as defined in Eq.(10). Then use the orthogonality relation of to show that , and that .
snlmm
snlmm
1126A.



(19) 11112222111002112117210232i..............

d) If you measure the energy, what are the possible outcomes and corresponding probabilities?

e) One of your classmates claims that since is a combination of different energy levels, the electron will spend part of its time in those levels and therefore i