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(PHYS234)[2008](s)midterm~ph_kpx^_10547.pdf
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PHYS234 Quantum Mechanics I Sample Midterm Examination

Formulae:
NOTE: not every formula listed here is needed in answering this exam paper
One-dimensional time-dependent Schr.dinger equation (1) One-dimensional time-independent Schr.dinger equation (2) Hamiltonian operator for a particle (3) First quantization rule: , (4) Canonical commutation rule (5) Eigenstates of momentum operator (plane waves): (6) Eigenstates of position operator : (7) An operator A is Hermitian if for any states and , (8) Mathematical formulae (9) , , Integration by parts 2222iVtmx............
222()2dVxEmdx.......
222()2dHVxmdx....
.xx.
.dpidx...
..[,]xpi..
.p
/12ipxpe....
.x
'(')xxx...
.
.
*AAA........
220104/5sin2cos21, 0!, 034iiiiaxnaxnxaeeieeedxaanxedxaaxedxa..............................
2220dykydx..
20k.
~ikxye..
2220dyydx...
20..
~xye...
udvuvvdu....

Question 1 [12 points] Consider the following one-dimensional potential,




a) Given that the lowest energy state of a particle in this potential has energy E1 as shown in the above diagram. Copy this diagram in your answer book, including the energy level E1, and sketch the lowest energy state wave function. Indicate the classically forbidden region and all important features such as values, functional forms, etc, and give your reason why it should have these features. There is no need to solve this problem exactly.

b) Repeat part a) for the first excited state whose energy is E2 as shown.


Question 2 [30 points] Consider a two dimensional Hilbert space. Wavefunctions are column vectors and basis vectors are and . The Hamiltonian H and two observables A and B are: , , . We denote the eigenvector of an operator Q with eigenvalue by . You are given that H has eigenvalues and , and eigenvectors , , and A has eigenvalues 1 and , and eigenvectors , . 10......
01......
00H..........
00iAi........
0110B.......
.
,Q.
.
..
1,0H........
0,1H.........
1.
11,12Ai.......
11,12Ai.........

a) You are told that if you measure the energy of a particular state, you have 36% of getting and 64% of getting. Write down its wavefunction as best as you can and explain why it must be so. Note that the measurement has not been performed.

b) Under what condition do we consider two wavefunctions to be identical?

c) If you measure A on a state with wavefunction , what is the probability of getting 1? Make sure your answer make sense. 34i........


d) If you measure A on and get 1, what is the wavefunction immediately after the measurement? .


e) At a time after the measurement in d), you measure B. What are the possible outcomes and probabilities? t....


f) Right after the measurement on B in e), but without knowing what the outcome is, A is immediately measured again. What is the probability of getting 1?


continue on next page
Question 3 [13 points] Two operators and are defined by , fo