1. Consider a particle of mass m that is free to move in a one-dimensional region of length L that closes on itself (for instance, a bead which slides frictionlessly on a circular wire of circumference L.

(a) Show carefully that the states |n> = (1/L)^1/2 exp(2pinx/L) and the energies E_n = 2 hbar² p² n² / (mL²).

(b) What is the time dependence of Y_n(x,t)?

(c) What values of n are allowed? What is the degeneracy of the state |n>?

(d) In state |n> of this system, what values of p are allowed, and what is the probability of a measurement of p^op giving each of these values?

2. Suppose that in the preceding problem, a term H' = b d(x - L/2) with b small is added to the Hamiltonian. Calculate to first order in the parameter b the energy of the state(s) whose unperturbed energy is 2 hbar² p² / (mL²)

3. (a) An electron is in a state with spin up along the z-axis, i.e.

    1
    0  .

    (b). If only electrons whose spin is up-along-x are kept in the apparatus after determining the spin direction, what is the wave function of the electron after the measurement?

    (c). What is the probability of now finding the electron in a state that has spin down along the z axis? Prove your answer.

For reference, the Pauli spin matrices for x, y, and z are

      0  1     0  -i     1   0
      1  0  ,  i   0  ,  0  -1

4. Let us invent a particle called the nothun, which has spin 1/2, feels a Coulomb ( - e² / 4 p e ₀ r) attraction to a proton, and interacts with any other nothun with a potential aS₁^.S₂ , where the parameter a is small. Two nothuns do not exert Coulomb forces on each other. The mass m of a nothun is small compared to the mass of a proton.

(a) Neglecting a, what is the properly symmetrized ground state of a system of 2 nothuns and a proton?

(b) What is the total spin S of the nothuns in this state?

(c) To first order in a, what is the energy of the ground state of this system?

5. (a) Show that y₁(x) = [4 m w/ (p hbar)]^1/4 x exp( - x² / 2) where x = (m w / hbar)^1/2 x is a solution for the one-dimensional harmonic oscillator with energy E = 3 h-bar w / 2 .

(b) Using the results of part (a), find the energy and wavefunction for the lowest state(s) in the three-dimensional harmonic oscillator with an angular momentum whose magnitude is h-bar (l=1). What is the degeneracy of this state?

1. Consider the potential

(a) Show carefully that the states |n> = (2/a)^1/2 sin(pnx/a) and the energies E_n = hbar²p²n²/a².

(b) What is the time dependence of Y_n(x,t)?

(c) What values of n are allowed? What is the degeneracy of the state |n>?

(d) Write down the matrix <n'|H|n> [at least enough of it that the form is apparent].

(e) In state |n> of this system, what values of p are allowed, and what is the probability of a measurement of p^op giving each of these values?

2. If in the preceding problem, a term H' = b cos(2px/a) with b small is added to the Hamiltonian, what is the energy of each state to first order in the parameter b? [Hint: 2 sin A sin B = cos(A+B) - cos(A-B) and 2 cos A cos B = cos(A+B) + cos(A-B) ].

3. Given that the operators     S_± = S_x ± iS_y     obey     S_±|s=1/2, s_z=m>   =   hbar |s=1/2, s_z=m±1>     whenever the state on the right-hand side of the relation exists, show that |s=1/2, s_z=1/2> is an eigenstate of the operator S² with eigenvalue (3/4)hbar² .