(a)  Explicitly find the expectation value of the potential energy for the first excited energy level of the one-dimensional harmonic oscillator and compare it to the total energy of this level.
(b)  Find the most general expression for the first excited state of the two-dimensional isotropic harmonic oscillator in terms of the eigenstates {|n>} of the 1-d oscillator.
(c)  Find the expectation values of x² and y² using the state vector from part (b).
What is the expectation value of U(x,y) = ½mω²(x² + y²)?

Find a linear combination of |+> and |-> kets that minimizes the uncertainty product (ΔS_x)²(ΔS_y)².  Verify explicitly, that for the product you found the uncertainty relation for S_x and S_y is not violated.

Consider the state space E = E₁⊗ E₂ of two non-identical spin 1/2 particles spanned by the basis vectors {|++>, |+->, |-+>, |-->}.   Find the common eigenvectors of S² and S_y.  Express these eigenvectors in terms of the basis vectors  {|++>, |+->, |-+>, |-->}.

The wave function Ψ(r) of a spinless particle is Ψ(r) = Nz²exp(-r²/b²), where b is a real constant and N is a normalization constant.
(a)  If L² is measured, what results can be obtained and with what probabilities?
(b)  If L_z is measured, what results can be obtained and with what probabilities?
(c)  Is Ψ(r) an eigenfunction of L² or L_z?

For the ground state of the one-dimensional harmonic oscillator, show that <p> = 0, <p²> = (ħ²/4)<x²>.

Solution:
Let X_s = (mω/ħ)^1/2X,  P_s =  (mωħ)^-1/2P, 
a = (2)^-1/2(X_s + iP_s), and  a^† = (2)^-1/2(X_s - iP_s),  [a,a^†] = 1.
Then X_s = (2)^-1/2(a^† + a),  P_S = i(2)^-1/2(a^† - a).
For a harmonic oscillator in its ground state  <X> = <P> = 0.

(ΔX)² = <X²> - <X>² = <X²>,  (ΔP)² = <P²> - <P>² = <P²>.
X² = (ħ/(2mω))(a^† + a)(a^† + a) = (ħ/(2mω))(aa + aa^† + a^†a + a^†a^†),
P² = -(mħω/2)(a^† - a)(a^† - a) = (mħω/2)(-aa + aa^† + a^†a - a^†a^†).
<Φ_n|a^†a^†|Φ_n> = <aΦ_n|a^†Φ_n> ∝ <Φ_n-1|Φ_n+1> = 0.
<Φ_n|aa|Φ_n> = <a^†Φ_n|aΦ_n> ∝ <Φ_n+1|Φ_n-1> = 0.
<Φ_n|a^†a|Φ_n> = <aΦ_n|aΦ_n> = n.
<Φ_n|aa^†|Φ_n> = <a^†Φ_n|a^†Φ_n> = n + 1.
<X²> = (ħ/(2mω))(2n + 1) = (ħ/(mω))(n + ½).
<P²> = (mħω/2)(2n + 1) = mħω(n + ½).
ΔXΔP = (n + ½)ħ.
ΔXΔP = ½ħ for n = 0, i.e. for the ground state.

Problem:

What are the energy levels of a particle of mass m moving in the one-dimensional potential well defined by U(x) = ∞ for x < 0, U(x) = ½kx² for x > 0?
At t = 0, an ensemble of particles is in the state |Ψ> = ¼|0> + ½|1> + (i√11/4)|2>,
where |0>, |1>, and |2> are the first three energy eigenstates of the linear harmonic oscillator.
(a)    Is |Ψ> normalized?
(b)    If a single observation is made on one of the particles, what likely values of energy will turn up, and with what probability?
(c)    If the experiment is repeated many times, each time on a different particle, what is the average value of the energy?
(d)    If a particle is left undisturbed, what is its state at time t?
(e)     Find <x(t)>.

Problem :

Two states of a spin 1/2 particle are represented in the eigenbasis of S_z by
|Ψ₁> =  (1/√2)|+> + (i/√2)|->,    |Ψ₂> =  (-i/√3)|+> + √(2/3)|->.
(a)  Find their representation in the eigenbasis of S_y.
(b)  Find the amplitude <Ψ₁|Ψ₂> in the S_z basis and show that this amplitude remains unchanged when calculated in the S_y basis. (Show your work.)
(c)   The Hamiltonian for the particle is H = ω₀S_z.  Find |Ψ₁(t)>.  At what times t is |Ψ₁(t)> an eigenvector of S_x?