A particle is represented (at time t = 0) by the wave function
ψ(x,t) = A(a2 - x2) if -a < x < a, ψ(x,t) = 0 otherwise.
(a) Determine the normalization constant A.
(b) What is the expectation value of x (at time t = 0)?
(c) What is the expectation value of p (at time t = 0)?
(d) Find the expectation value of x2.
(e) Find the expectation value of p2.
(f) Find the uncertainty in x (Δx).
(g) Find the uncertainty in p (Δp).
(h) Check that your results are consistent with the uncertainty principle.
Solution:
An operator A, representing observable A, has two normalized eigenstates
Ψ1 and Ψ2,
with eigenvalues a1 and a2, respectively. An operator
B, representing observable B, has
two normalized eigenstates Φ1 and Φ2, with eigenvalues b1
and b2, respectively. The eigenstates
are related by
Ψ1 = (3Φ1 + 4Φ2)/5,
Ψ2 = (4Φ1 - 3Φ2)/5.
(a) Observable A is measured, and the value a1 is obtained. What is the state of
the system immediately after this measurement?
(b) If B is now measured immediately following the A measurement, what are the possible
results, and what is the probability of measuring each result?
(c) Right after the measurement of B, A is measured again. What is the
probability of getting a1?
Solution:
(a) Show that the eigenvalues of a general 2×2 matrix A can be expressed as
λ±= ½T ±
(¼T2 - D)½,
where D is the determinant of A and T is the trace of A (sum of diagonal
elements).
Show that the matrix
A =
0 | m | ||
m | M |
with M >> m has two eigenvalues, with one much larger than the other.
(b) Show that the most general form of a 2×2 unitary matrix U with unit determinant can be parameterized as
U =  
a | b | ||
-b* | a* |
,
subject to the constraint aa* + bb* = 1, where * denotes complex conjugation.
Solution:
A =  
a | b | ||
c | d |
we have
a - λ | b | ||
c | d - λ |
= 0.
A =  
0 | m | ||
m | M |
and assuming M >> m gives T = M, D =
-m2,
λ± = ½M ± (¼M2 + m2) = ½M ± ½M(1 + 4m2/M2)½
≈ ½M ± ½M(1 + 2m2/M2)
= ½M ± (½M + m2/M).
The eigenvalues therefore are λ+ ≈ M, λ- ≈ m2/M,
|λ+| >> |λ-|.
(b) Parameterize the matrix in terms of complex parameters a, b, c, and d, and
require the unitarity condition UU† = I.
U =  
a | b | ||
c | d |
U† =  
a* | c* | ||
b* | d* |
.
UU† = I --> aa* + bb*
= 1, cc*
+ dd* = 1, ac* + bd* =
0.
b = -c*a/d*.
det(U) = 1 --> ad - bc = 1, ad + |c|2a/d* = 1, a(|d|2
+ |c|2) = d*, a = d*, b = -c*.
Therefore
U =  
a | b | ||
-b* | a* |
|a|2 + |b|2 = 1.
Consider a three-state system. In some orthonormal basis, {|1>, |2>, |3>}, the matrix of the Hamiltonian operator H is
H =
0 | 2 | 0 | ||
2 | 0 | 0 | ||
0 | 0 | 1 |
.
Choose units such that ħ = 1. Find the matrix of the evolution operator in the same basis.
Solution:
-E | 2 | 0 | ||
2 | -E | 0 | ||
0 | 0 | 1-E |
= 0.
E2(1 - E) - 4(1 - E) = 0. The solutions to this equation are E1
= -2, E2 = 1, E3 = 2.
The corresponding
normalized eigenvectors are
|E1> = 2-½
-1 | ||
1 | ||
0 |
, |E2> =
0 | ||
0 | ||
1 |
, |E3> = 2-½
1 | ||
1 | ||
0 |
.
The matrix of H in its eigenbasis, with the eigenvectors ordered as {|E1>, |E2>, |E3>}, is
H =
-2 | 0 | 0 | ||
0 | 1 | 0 | ||
0 | 0 | 2 |
.
The matrix of the evolution operator in the eigenbasis of H is
U' = exp(-iHt) =
exp(i2t) | 0 | 0 | ||
0 | exp(-it) | 0 | ||
0 | 0 | exp(-i2t) |
.
The matrix of the evolution operator in the original {|1>, |2>, |3>} basis is
U= DU'D†, where D is the unitary transformation from the
{|1>, |2>, |3>} basis to the {|E1>, |E2>, |E3>} basis.
The matrix of D is
D =
-2-½ | 0 | 2-½ | ||
2-½ | 0 | 2-½ | ||
0 | 1 | 0 |
.
The matrix of D† is
D† =
-2-½ | 2-½ | 0 | ||
0 | 0 | 1 | ||
2-½ | 2-½ | 0 |
.
Therefore the matrix of the evolution operator in the original {|1>, |2>, |3>} basis is
U =
cos(2t) | -isin(2t) | 0 | ||
-isin(2t) | cos(2t) | 0 | ||
0 | 0 | exp(-it) |
.
Consider a two state system with kets |0> and |1> defining a set which is
assumed to be orthonormal and complete.
The Hamiltonian of the system is H = ħωa†a, where the operators a and a†are defined by
a†|0> = |1>, a|1> = |0>, a†|1> = a|0> = 0.
(a) Find the eigenvalues of H.
(b) Write down the Dirac (bra - ket) representation of the operators H, a, and
a†.
(c) What are the anti-commutators of the operators a and a†?
Solution: