G25.2666: Quantum Mechanics II

Problem set #3
Due: March 1, 2005

1.

Consider three spin-1/2 particles with spins ${\bf S}_1$, ${\bf S}_2$ and ${\bf S}_3$. Define ${\bf S}={\bf S}_1+{\bf S}_2+{\bf S}_3$ as the total spin. Find a set of eigenvectors for $S^2$ and $S_z$ in terms of the direct products of the spin eigenstates, $\vert 1/2\;\; m_{s_i}\rangle$, $i=1,2,3$, of the individual particles.

2.

Consider a system composed of two spin-1/2 particles fixed in space described by a Hamiltonian of the form

$$H = \omega_1 S_{1z} + \omega_2 S_{2z}$$

where $S_{1z}$ and $S_{2z}$ are the $z$-components of the spin operators ${\bf S}_1$ and ${\bf S}_2$ for the two particles, respectively.

a.

Suppose the initial state vector is a triplet state

$$\vert\Psi(0)\rangle = {1 \over \sqrt{2}} \left[\vert 1/2;-1/2\rangle + \vert-1/2;1/2\rangle\right]$$

At time, $t$, ${\bf S}^2 = ({\bf S}_1+{\bf S}_2)^2$ is measured. What values can be found and with what probabilities?

b.

Suppose the initial state vector is the singlet state

$$\vert\Psi(0)\rangle = {1 \over \sqrt{2}} \left[\vert 1/2;-1/2\rangle - \vert-1/2;1/2\rangle\right]$$

At time, $t$, ${\bf S}^2 = ({\bf S}_1+{\bf S}_2)^2$ is measured. What values can be found and with what probabilities?

c.

Can this Hamiltonian ever evolve a triplet state into a singlet state at any time $t$? Can you write down a Hamiltonian that can effect such an evolution?

3.

Consider a single quantum mechanical particle in one dimension with position and momentum operators $X$ and $P$, respectively. The Hamiltonian of the system is

$$H = {P^2 \over 2m} + V(X)$$

Recall that the time evolution of the system can be studied equally well using a stationary state picture (the Heisenberg picture) in which the operators $P$ and $X$ evolve in time accoring to Heisenberg's equations of motion:

$${dX \over dt} = {1 \over i\hbar}[X,H]$$

$${dP \over dt} = {1 \over i\hbar}[P,H]$$

The general solution to Heisenberg's equations will be

$$X(t;X,P) = e^{iHt/\hbar}Xe^{-iHt/\hbar}$$

$$P(t;X,P) = e^{iHt/\hbar}Pe^{-iHt/\hbar}$$

where $X$ and $P$ are the usual Schrödinger operators for position and momentum, respectively.

Consider a set of Hermitian operator-valued functions $\{T_k(X,P)\}$ in terms of the Schrödinger operators, $X$ and $P$ that satisfy a Lie algebra:

$$\left[T_k(X,P),T_l(X,P)\right] = \sum_m C_{klm}T_m(X,P)$$

where $C_{klm}$ are the structure constants of the group and are, therefore, assumed known. Such a set of functions forms an operator basis in terms of which any Hermitian operator-valued function $F(X,P)$ can be expanded according to:

$$F(X,P) = \sum_k f_k T_k(X,P)$$

where $f_k$ is a set of complex expansion coefficients.

a.

Express the Hamiltonian and solutions to Heisenberg's equations of motion as expansions with respect to the operator functions $T_k(X,P)$. Which sets of expansion coefficients will be functions of time?

b.

Starting from Heisenberg's equations, derive equations of motion for the time-dependent coefficients in terms of the structure constants of the Lie group.

c.

What are the initial conditions on your equations of motion?

d.

Of the conditions:

$$X^{\dagger}(t;X,P) = X(t;X,P)$$

$$P^{\dagger}(t;X,P) = P(t;X,P)$$

$$\left[X(t;X,P),P(t;X,P)\right] = i\hbar I$$

which of these impose additional conditions on the equations of motion and what, if any, are these conditions?