A particle is represented (at time t = 0) by the wave function

ψ(x,t) = A(a^{2} - x^{2}) 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 x^{2}.

(e) Find the expectation value of p^{2}.

(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.

An operator A, representing observable A, has two normalized eigenstates
Ψ_{1} and Ψ_{2},
with eigenvalues a_{1} and a_{2}, respectively. An operator
B, representing observable B, has
two normalized eigenstates Φ_{1} and Φ_{2}, with eigenvalues b_{1}
and b_{2}, respectively. The eigenstates
are related by

Ψ_{1} = (3Φ_{1} + 4Φ_{2})/5,

Ψ_{2} = (4Φ_{1} - 3Φ_{2})/5.

(a) Observable A is measured, and the value a_{1} 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 a_{1}?

(a) Show that the eigenvalues of a general 2×2 matrix A can be expressed as

λ_{±}= ½T ±
(¼T^{2} - 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.

The corresponding function of an operator Ω is defined as F(Ω) = ∑

Let |Φ

F(Ω)|Φ

|Φ

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.

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^{†}?