P113A Quantum Mechanics Midterm 1, UCI (Undergraduate)

Physics 113AExam 1

May 5, 2009

June 6, 2009

For question 1, 2, 3, consider a particle of mass m that is one-dimensionalpotential that goes to infinity at plus and minus infinity. The solutions tothe time-independent Schrodinger Equation are indicated by ψn(x) = |n >for n = 1, 2, 3, · · ·. The energy of each state in terms of the ground state E1

is given by E1n3.

1) At time t = 0, the particle is equally likely to have an energy E2 andE4 and only one of those two energies, write an expression for the time-independent, normalized wavefunction in terms of ψn(x) = |n > and theground energy E1. (2pts)

3) Given the state in question 1, do you expect probability of finding theparticle in a range of 0 < x < a to be time-dependent? Make sure to brieflyjustify your answer. (3 pts)

4) Consider a two-dimensional vector space with basis |1 > and |2 >. Writea matrix representation of the operator T that converts |1 >→ −ı|2 > and|2 >→ ı|1 >. In terms of the vectors |1 > and |2 >, what two normalizedvectors form the basis in which T is diagonal? (6 pts)

5) Consider a particle of mass m moving in a harmonic oscillator poten-tial with natural frequency ω. The (orthonormal) stationary states are in-dicated by ψn(x) = |n >, with n = 0, 1, 2, · · ·. Consider the state ψ(x, 0) =

1√2(ψ0(x) + ψ1(x)). Compute < x >, and < p > as functions of times using

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any methods you prefer. (6 pts)

6) Consider a particle which is equally likely to be anywhere in the rangefrom 0 < x < a. Now consider two differents potentials that are definedusing the same zero point.

Potential 1:V1(x) =1

2k(x− a/2)2

Potential 2:V2(x) =

∞ and x ≤ 0−V0 and 0 < x ≤ a∞ and x > a

In which potential will the particle have the greater expectation value for itsenergy? Make sure to justify your answer, but there is no need to computespecific expressions for the expectation values. (5 pts)

7) Consider a particle in a potential given by V (x) = − h2

maδ(x), where a = 1

in the appropriate units of length. If at time t = 0, the particle is in thestate (again, where the number are in the appropriate units of length)

ψ(x, 0) =

0 if x < −1√32(1 + x) if −1 < x < 0√

32(1− x) if 0 < x < 1

0 if x > 1

,

find the probability of measuring an energy of E = − h2

2ma2 , where a = 1 inthe appropriate units of length. (5 pts)

8) Consider a particle in the following potential:

V (x) =

∞ for x ≤ −a−V0 for −a < x ≤ a∞ for x > a

.

Find all expressions for all of the bound states’ energies and wavefunctionsas functions of position, (Hint: consider the symmetric and anti-symmetricsolutions separately.) (10 pts)

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