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The HET Group
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HOMEWORK 3
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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., March 22, 2011
Problem 3.1
A particle of mass $m$ is placed in an infinite one-dimensional well of width $a$,
\[ V(x) = \left\{ \begin{array}{lll} 0 & , & 0\le x \le a \\ +\infty & , & x\lt 0 \ \ \mathrm{or} \ \ x\gt a \end{array} \right. \]
It is subject to a perturbation
\[ W(x) = aw_0 \delta \left( x - \frac{a}{2} \right) \]
- Calculate the first-order change in the energy levels due to $W$.
- Solve the problem exactly. Show that the energy levels are solutions of one of the two equations
\[ \sin \frac{ka}{2} = 0 \ \ \ \ \mathrm{or} \ \ \ \ \cot \frac{ka}{2} = - \frac{maw_0}{\hbar^2 k} \]
where $k = \sqrt{2mE}/\hbar$.
Solve for the energy levels graphically and find explicit expressions in the limits $w_0\to\infty$ and $w_0\to 0$ (the latter should agree with your result in part a.).
Problem 3.2
A particle of mass $m$ is placed in an infinite two-dimensional well of width $a$,
\[ V(x,y) = \left\{ \begin{array}{lll} 0 & , & 0\le x \le a \ \ \mathrm{and} \ \ 0\le y \le a \\ +\infty & , & \mathrm{everywhere~else} \end{array} \right. \]
It is subject to a perturbation
\[ W(x,y) = \left\{ \begin{array}{lll} w_0 & , & 0\le x \le \frac{a}{2} \ \ \mathrm{and} \ \ 0\le y \le \frac{a}{2} \\ 0 & , & \mathrm{everywhere~else} \end{array} \right. \]
- Calculate the first-order change in the ground state energy due to $W$.
- Calculate the first-order change in the first excited energy level due to $W$.
Find the corresponding eigenstates to zeroth order in $w_0$.
Problem 3.3
Consider a particle of mass $\mu$ constrained to move on a circle of radius $R$. The wavefunction is a function of the angle $\theta$ only and periodic,
\[ \psi (\theta + 2\pi) = \psi(\theta) \]
It is normalized by
\[ \int_0^{2\pi} d\theta |\psi(\theta)|^2 = 1 \]
- Show that the operator
\[ M = -i\hbar \frac{d}{d\theta} \]
is hermitian.
What is its physical meaning, eigenvalues and corresponding normalized eigenfunctions?
- The kinetic energy of the particle is
\[ H_0 = \frac{M^2}{2\mu R^2} \]
Calculate the eigenvalues and corresponding eigenfunctions of $H_0$.
Are the energy levels degenerate?
- At time $t=0$, the wavefunction of the particle is
\[ \psi (\theta) = N \cos^2\theta \]
Find the wavefunction at subsequent time $t\gt 0$.
What is the probability that the particle will be found between $\theta$ and $\theta+d\theta$ at time $t\gt 0$?
- Assume that the particle has charge $q$ and a uniform electric field $\mathcal{E}$ is switched on in the $\theta=0$ direction.
We must therefore add to the Hamiltonian $H_0$ a perturbation,
\[ W = -q\mathcal{E} R\cos\theta \]
Calculate the new wave function of the ground state to first order in $\mathcal{E}$ and the electric susceptibility $\chi_e$.
- For the ethane molecule, $CH_3 - CH_3$, consider the rotation of one $CH_3$ group relative to the other about a straight line joining the two carbon atoms.
To account for the electrostatic interaction between the two $CH_3$ groups, we
add to $H_0$ a perturbation
\[ W = b \cos (3\theta) \]
Calculate the energy of the new ground state to second order in $b$ and the corresponding wavefunction to first order in $b$.
Problem 3.4
Consider a system of angular momentum $J=1$. The Hamiltonian is
\[ H_0 = a J_z + \frac{b}{\hbar} J_z^2 \]
where $a,b\gt 0$.
- What are the energy levels of the system?
For what value of the ratio $b/a$ is there degeneracy?
- The magnetic moment of the system is
\[ \vec\mu = \gamma \vec J \]
where $\gamma \lt 0$ is the gyromagnetic ratio.
A uniform magnetic field $\vec B = B_0 \hat n\ $ is applied in a direction with polar angles $\theta$ and $\phi$.
Therefore, we need to add to $H_0$ a perturbation
\[ W = \omega_0 \vec J \cdot \hat n \]
where $\omega_0 = -\gamma B_0$ is the Larmor angular frequency.
Write $W$ as a $3\times 3$ matrix.
- Assume $b=a$ and $\hat n = \hat x$.
Calculate the energy levels to first order in $\omega_0$ and the corresponding eigenstates to zeroth order in $\omega_0$.
- Now assume that $b=2a$ and $\hat n$ points in an arbitrary direction.
Calculate the ground state of $H_0+W$ to first order in $\omega_0$.
Calculate the mean value $\langle \vec\mu\rangle$ of the magnetic moment in the ground state and deduce the $3\times 3$ magnetic susceptibility matrix defined by
\[ \langle \mu_i \rangle = \sum_{j=1}^3 \chi_{ij} B_j \]
Is $\langle\vec\mu\rangle$ parallel to $\vec B$?
Problem 3.5
Consider a nucleus of spin $I = 3/2$ situated at the origin ($\vec r = \vec 0$).
It is subject to a non-uniform electric field
\[ \vec E = - \vec\nabla U \ \ \ \ \mathrm{where} \ \ \ \ \nabla^2 U = 0 \]
Furthermore, we choose axes so that at the origin
\[ \frac{\partial^2 U}{\partial x \partial y} =
\frac{\partial^2 U}{\partial y \partial z} =
\frac{\partial^2 U}{\partial z \partial x} = 0 \]
The Hamiltonian is due to the interaction between the gradient of the electric field at the nucleus (origin) and the electric quadrupole moment of the nucleus.
It reads
\[ H_0 = \frac{qQ}{2\hbar^2 I(2I-1)} \left[ a_x I_x^2 + a_y I_y^2 + a_z I_z^2 \right] \]
where $q$ is the charge of an electron, the constant $Q$ is proportional to the quadrupole moment of the nucleus and
\[ a_x = \left. \frac{\partial^2 U}{\partial x^2} \right|_{\vec r = \vec 0} \ \ , \ \ \ \
a_y = \left. \frac{\partial^2 U}{\partial y^2} \right|_{\vec r = \vec 0} \ \ , \ \ \ \
a_z = \left. \frac{\partial^2 U}{\partial z^2} \right|_{\vec r = \vec 0} \]
- Assuming cylindrical symmetry (under rotations about the $z$-axis), show that
\[ H_0 = A \left[ 3I_z^2 - I(I+1) \hbar^2 \right] \]
and find the constant $A$.
What are the eigenvalues of $H_0$, their degrees of degeneracy and corresponding eigenvectors?
- Show that in the general case,
\[ H_0 = A \left[ 3I_z^2 - I(I+1) \hbar^2 \right] + B \left[ I_+^2 + I_-^2 \right] \]
and find the constants $A$ and $B$.
What are the eigenvalues of $H_0$, their degrees of degeneracy and corresponding eigenvectors?
[HINT: Write $H_0$ as a $4\times 4$ matrix and show that it splits into two $2\times 2$ submatrices.]
- Switch on a uniform magnetic field $\vec B = B_0 \hat n$ (in addition to the electric field).
The nucleus interacts with it because it has magnetic moment
\[ \vec\mu = \gamma \vec I \]
where $\gamma$ is the gyromagnetic ratio. Set $\omega_0 = - \gamma B_0$.
What term $W$ must be added to $H_0$ to account for the interaction with the magnetic field?
Calculate the new energy levels of the system to first order in $\omega_0$.
- Suppose $\vec B$ is along the $z$-axis ($\hat n = \hat z$).
What are the Bohr frequencies which can appear in the evolution of $\langle I_x \rangle$?
Deduce the shape of the NMR spectrum which can be observed with a radiofrequency field oscillating in the $x$-direction.
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Department of Physics and Astronomy