Using the definition of angular momentum**L** = **r** × **p**,

(a) show that [L_{x}, L_{y}] = iħ L_{z},_{
} [L_{y}, L_{z}] = iħ L_{x},_{ } [L_{z},
L_{x}] = iħ L_{y}.

(b) Using this result show that L^{2}
commutes with L_{z}.

A system has a wave function ψ(x,y,z) = N*(x + y + z)*exp(-r^{2}/α^{2})
with α real. If L_{z} and L^{2} are measured, what are the
probabilities of finding 0 and 2ħ^{2}?

Consider the Hamiltonian of a spinless particle of charge q_{e} in
the presence of a static and uniform magnetic field **B** = B **k**.

H
= (1/(2m)) (**p** - q_{e}**A**(**r**,t))^{2}.

By
using the gauge in which **∇**∙**A** = 0, demonstrate that the Hamiltonian
can be expressed as

H = p^{2}/(2m) - (q_{e}/(2m)) **L**∙**B**
+ (q_{e}^{2}B^{2}/(8m))(x^{2} + y^{2}).

Note that the second term corresponds to the linear coupling between the
external field and the magnetic moment.

An electron is a rest in magnetic field **B** = B_{0} **k** + B_{1}(cos(ωt)
**i** + sin(ωt) **j**). B_{0}, B_{1}, and ω are
constants. At t = 0 the electron is in the |+> eigenstate of S_{z}.
Let **μ** = -γ**S** be the magnetic moment of the electron and ω_{0}
= γB_{0}, ω_{1} = γB_{1} be constants.

(a)
Construct the Hamiltonian matrix for this system and write down the
time-dependent Schroedinger equation,

iħ

∂α/∂t | ||

∂β/∂t |

= H(t)

α | ||

β |

.

in matrix form in the {|+>, |->} basis.

(b) Convert this equation into a "Schroedinger equation"
with a time independent "Hamiltonian" by choosing new expansion coefficients

a(t) = exp(iωt/2)α(t), b(t) = exp(-iωt/2)β(t).

(Hint: Given
∂α/∂t and ∂β/∂t find ∂a/∂t and ∂b/∂t.)

(c) Find the eigenvalues and
eigenvectors if the "Hamiltonian" in part (b).

Hint: Write

H' = A

cosθ | sinθ | ||

sinθ | -cosθ |

.

The eigenvalues of the matrix are λ = ±1, and the
corresponding eigenvectors are

|ψ_{+}> = cos(θ/2)|+> +
sin(θ/2)|->, |ψ_{-}> = -sin(θ/2)|+> + cos(θ/2)|->.

(d)
The Schroedinger equation now implies that

U(t,0)|ψ_{+}> = |ψ_{+}>exp(-iAλ_{+}t/ħ),
and U(t,0)|ψ_{-}> = |ψ_{-}>exp(-iAλ_{-}t/ħ).

Find the
probability of finding the electron in the |-> eigenstate of S_{z} as a
function of time.

For any two quantum-mechanical operators A and B, the uncertainty principle
says that

<(ΔA)^{2}><(ΔB)^{2}> ≥ ¼|<[A,B]>|^{2}.
Consider a spin ½ particle. Show that for the spin operators S_{x}
and S_{y} the eigenstate |+> of the S_{z} operator is a minimum
uncertainty state.