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The HET Group
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HOMEWORK 2
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TEXTBOOKS:
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Quantum Mechanics, Vol. 2, by Claude Cohen-Tannoudji, et al.,
Wiley-VCH.
Quantum Mechanics with Basic Field Theory, by Bipin R. Desai, Cambridge.
Quantum Mechanics, by Eugen Merzbacher, Hamilton.
Modern Quantum Mechanics, by J. J. Sakurai, Addison-Wesley.
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due date: Tue., February 22, 2011
Problem 2.1
The deuterium atom consists of a nucleus of spin $I=1$ and an electron.
The electronic angular momentum is $\vec J = \vec L + \vec S$, where $\vec L\ $
is the orbital angular momentum of the electron and $\vec S$ is its spin.
The total angular momentum of the atom is $\vec F = \vec J + \vec I$, where $\vec I$ is the nuclear spin.
The eigenvalues of $\vec J^2$ and $\vec F^2$ are $J(J+1)\hbar^2$ and $F(F+1)\hbar^2$, respectively.
- What are the possible values of the quantum numbers $J$ and $F\ $ for a deuterium atom in the $1s$ ground state?
- What are the possible values of the quantum numbers $J$ and $F\ $ for a deuterium atom in the $2p$ excited state?
Problem 2.2
The hydrogen atom consists of a nucleus which is a proton of spin $I=1/2$ and an electron.
The electronic angular momentum is $\vec J = \vec L + \vec S$, where $\vec L\ $
is the orbital angular momentum of the electron and $\vec S$ is its spin.
The total angular momentum of the atom is $\vec F = \vec J + \vec I$, where $\vec I$ is the nuclear spin.
The eigenvalues of $\vec J^2$ and $\vec F^2$ are $J(J+1)\hbar^2$ and $F(F+1)\hbar^2$, respectively.
- What are the possible values of the quantum numbers $J$ and $F\ $ for a hydrogen atom in the $2p$ excited state?
- The magnetic moment of the electron is
\[ \vec\mu = \frac{\mu_B}{\hbar} [ \vec L + 2\vec S ] \]
For fixed $J$ and $F$ explain why we have
\[ \vec\mu = g_{JF} \frac{\mu_B}{\hbar} \vec F \]
and calculate the possible values of the Lande factors $g_{JF}$ in the $2p$ level.
Problem 2.3
Consider a system consisting of two spin 1/2 particles whose orbital variables are ignored.
The Hamiltonian of the system is
\[ H = \omega_1 S_{1z} + \omega_2 S_{2z} \]
where $\omega_1$ and $\omega_2$ are given (real) constants.
Let $\vec S = \vec S_1 + \vec S_2$ be the total spin of the system.
- At $t=0$ the system is in the state
\[ |\psi (0) \rangle = \frac{1}{\sqrt{2}} [ |+-\rangle + |-+\rangle ] \]
If at time $t\gt 0$, $\vec S^2$ is measured, what are the possible outcomes and with what probability will each occur?
- If the initial state is
\[ |\psi (0) \rangle = \alpha |++\rangle + \beta |+-\rangle + \gamma |-+\rangle + \delta |--\rangle \]
calculate $\langle \vec S^2 \rangle$ and $\langle S_x \rangle$ at $t\gt 0$.
What Bohr frequencies can appear in the evolution of $\langle \vec S^2 \rangle$ and $\langle S_x \rangle$, respectively?
Problem 2.4
Consider two spin 1/2 particles of spins $\vec S_1$, $\vec S_2$, position vectors $\vec r_1$, $\vec r_2$ and masses $m_1$ and $m_2$, respectively.
Assume that the interaction between the two spinors is of the form
\[ W = U(r) + V(r) \frac{\vec S_1\cdot \vec S_2}{\hbar^2} \]
where $U$ and $V$ are given functions of the relative distance $r = |\vec r_2 - \vec r_1|$.
Let $\vec S = \vec S_1 + \vec S_2$ be the total spin of the system.
- Show that
\[ P_1 = \frac{3}{4} \mathbb{I} + \frac{\vec S_1\cdot \vec S_2}{\hbar^2} \ \ , \ \ \ \
P_0 = \frac{1}{4} \mathbb{I} - \frac{\vec S_1\cdot \vec S_2}{\hbar^2} \]
are projection operators onto the $S=1$ and $S=0$ states, respectively.
Deduce that the interaction can be written as
\[ W = W_1(r) P_1 + W_0(r) P_0 \]
and find $W_1$ and $W_0$.
- Consider motion in the center of mass frame in which the Hamiltonian reads
\[ H = \frac{\vec p^2}{2\mu} + W \]
where $\mu = \frac{m_1m_2}{m_1+m_2}$ is the reduced mass.
Show that $H$ commutes with $\vec S^2$ and does not depend on $S_z$.
Hence show that for a given energy level $E$, the corresponding general eigenstate is of the form
\[ |E\rangle = \alpha \phi_0 (r) |00\rangle + \phi_1(r) [ \beta |11\rangle + \gamma |10\rangle + \delta |1-1\rangle ] \]
Find the eigenvalue equations obeyed by the wavefunctions $\phi_0$ and $\phi_1$.
Problem 2.5
For a vector operator $\vec V = ( V_x, V_y, V_z)$, define
\[ V_\pm = V_x \pm iV_y \]
\[ V_1 = - \frac{1}{\sqrt{2}} V_+ \ \ , \ \ \ \ V_0 = V_z \ \ , \ \ \ \ V_{-1} = \frac{1}{\sqrt{2}} V_- \]
For two vector operators, $\vec V\ $ and $\vec W$, define
\[ [ V \otimes W ]_M^{(K)} = \sum_p \sum_q \langle 1,1;p,q|K,M\rangle V_p W_q \]
where $p,q=-1,0,+1$ and $\langle 1,1; p,q|K,M\rangle$ are Clebsch-Gordan coefficients.
- Show that
\[ [ V\otimes W]_0^{(0)} = \lambda \vec V\cdot \vec W \]
and find $\lambda$.
- Express each of the three operators $[V\otimes W]_M^{(1)}$ ($M=-1,0,1$) in terms of the three components $(\vec V\times \vec W)_M$.
- Express the five operators $[V\otimes W]_M^{(2)}$ ($M=-2,-1,0,1,2$) in terms of $V_z$, $V_\pm$, $W_z$, $W_\pm$.
- Let $\vec V = \vec W = \vec R$, where $\vec R$ is the position operator of a given particle.
Express each of the five operators $[R\otimes R]_M^{(2)}$ in terms of the five components of the electric quadrupole moment operator $Q_2^M$ of the particle,
\begin{eqnarray}
Q_2^{\pm 2} &=& \frac{\sqrt{6}}{4} q (X\pm iY)^2 \\
Q_2^{\pm 1} &=& \mp \frac{\sqrt{6}}{2} q (X\pm iY)Z \\
Q_2^0 &=& \frac{1}{2} q (2Z^2 - X^2 -Y^2)
\end{eqnarray}
- Let $\vec V = \vec W = \vec L$, where $\vec L$ is the orbital angular momentum of a given particle.
Express each of the five operators $[L\otimes L]_M^{(2)}$ in terms of $L_z$, $L_\pm$.
Hence show that the matrix element
\[ \langle \ell m | [L\otimes L]_M^{(2)} |\ell' m'\rangle \]
is non-zero only if $\ell = \ell'$ and $m=m'+M$.
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Department of Physics and Astronomy