Consider a particle in one-dimension in a quantum state described by the wave function
ψ(x) = (3/a3)½(a - x) for 0 ≤ x ≤ a,
ψ(x) = 0 everywhere else.
(a) Where is the most probable place to find the particle?
(b)
What is the expected value for the position <x>?
Solution:
Consider a two-state system governed by the Hamiltonian H with energy eigenstates |E1> and |E2>,
where H|E1> = E1|E1> and H|E2> = E2
|E2>. Consider also two other states,
|x> = (|E1> + |E2>)/√2, and |y> = (|E1>
- |E2>)/√2.
At time t = 0 the system is in state |x>. At what
subsequent times is the probability of finding the system in state |y> the
largest, and what is that probability?
Solution:
Consider a free particle which is described at t = 0 by the normalized
Gaussian wave function
ψ(x,0) = (2a/π)¼exp(-ax2).
(a) Find the probability density |ψ(x,0)|2 of the particle.
(b) Find the Fourier transform Φ(k,0) of the wave function and the probability
density
|Φ(k,0)|2 in k-space.
Solution:
Is the ground state of the infinite square well an eigenfunction of the momentum
operator?
If so, what is the eigenvalue of the momentum of the particle in that state?
If not, why not?
Solution:
Consider a 2 by 2 matrix M with matrix elements m11,
m12, m21, and m22.
(a) What relationships between these matrix elements must exist for the matrix
M to be a unitary matrix, M = U?
(b) Show that the determinant of M = U has magnitude 1.
Solution:
U =
m11 | m12 | ||
m21 | m22 |
. U† =
m11* | m21* | ||
m12* | m22* |
.
UU† = I -->
|m11|2 + |m12|2 = 1, |m21|2
+ |m22|2 = 1, m11 m21* +
m12 m22* = 0.
U†U = I --> |m11|2
+ |m21|2 = 1, |m12|2 + |m22|2
= 1, m22 m21* + m12 m11*
= 0.
|m11|2 + |m12|2 = |m11|2
+ |m21|2 = 1 implies
|m21| = |m21|.
Since |m11|2
+ |m12|2 = 1 there exists an angle θ such that |m11|=
cosθ, |m12| = |m21| = sinθ.
|m11|2 + |m12|2 = |m21|2
+ |m22|2 = 1 implies
|m11| = | m22|, |m22|= cosθ.
Let mii = |mii|exp(iφii). Then
m11 m21* +
m12 m22* = 0 and m22 m21* + m12 m11*
= 0
implies φ11 - φ21 = φ12 - φ22 ± π
and φ22 - φ21 = φ12 - φ11
± π,
φ11 = -φ22 + φ21 + φ12 ± π.
If we choose the φ11 = -φ22, by factoring out a common
factor exp(iΔ) from all matrix elements then
φ21 = -φ12 ± π.
Therefore, except for some common factor exp(iΔ), we have
m11 = cosθ exp(iφ1), m12 = sinθ exp(iφ2),
m21 = -sinθ exp(-iφ2), m22 = cosθ exp(-iφ1).
(b)
det(U) = exp(iΔ)(cos2θ + sin2θ) = exp(iΔ).
|det(U)| = 1.