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The HET Group
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HOMEWORK 4
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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., April 19, 2011
Problem 4.1
Consider a one-dimensional harmonic oscillator of mass $m$, angular frequency $\omega_0$ and charge $q$.
Let $|n\rangle$ and $E_n = (n+ \frac{1}{2} ) \hbar\omega_0$ be the eigenstates and corresponding eigenvalues of its Hamiltonian $H_0$.
For $t\lt 0$, the oscillator is in the ground state $|0\rangle$.
At $t=0$, it is subjected to an electric field pulse of duration $\tau$.
The corresponding perturbation can be written as
\[ W(t) = \left\{ \begin{array}{ccl} -q\mathcal{E} x & , & 0\le t \le \tau \\
0 & , & t\lt 0 ~~~\mathrm{or}~~~ t\gt \tau \end{array} \right. \]
Let $\mathcal{P}_{0n}$ be the probability of finding the oscillator in the state $|n\rangle$ after the pulse.
- Calculate $\mathcal{P}_{01}$ using first-order time-dependent perturbation
theory. How does it vary with $\tau$ for fixed $\omega_0$?
- Calculate $\mathcal{P}_{02}$ using second-order time-dependent perturbation theory.
- Find the exact eigenvalues of the Hamiltonian $H_0+W$ and corresponding eigenstates (you may find it helpful to use the translation operator $e^{-iap/\hbar}$, where $a$ is a length parameter to be determined).
Deduce exact expressions for $\mathcal{P}_{01}$ and $\mathcal{P}_{02}$ (no need to sum the series). By expanding in $\mathcal{E}$, confirm the results of parts a. and b.
Problem 4.2
Consider two spin 1/2 particles coupled by an interaction of the form
\[ W = a(t) \vec S_1 \cdot \vec S_2 \]
where $a(t)\to 0$ as $t\to \pm\infty$ and the integral
\[ \mathcal{A} = \int_{-\infty}^{+\infty} a(t) dt \]
is finite.
- If at $t=-\infty$ the system is in the state $|+-\rangle$, calculate the state of the system at $t=+\infty$.
Find the probability $\mathcal{P} (+- \longrightarrow -+)$ of finding the system in the state $|-+\rangle$ at $t=+\infty$ in terms of $\mathcal{A}$.
- Calculate $\mathcal{P} (+-\longrightarrow -+)$ using first-order time-dependent perturbation theory.
Compare your result to the exact expression found in part a. and deduce the conditions for the validity of the approximation required for first-order perturbation theory.
- Now assume that the two spinors are also interacting with a uniform mangetic field in the $z$-direction, $\vec B = B_0 \hat z$.
The corresponding Hamiltonian is
\[ H_0 = - B_0 ( \gamma_1 S_{1z} + \gamma_2 S_{2z} ) \]
where $\gamma_1 \ne \gamma_2$.
Assuming
\[ a(t) = a_0 e^{-t^2/\tau^2} \]
calculate $\mathcal{P} (+-\longrightarrow -+)$ using first-order time-dependent perturbation theory. Plot this probability as a function of $B_0$ for fixed $a_0$ and $\tau$.
Problem 4.3
A simple model of the photoelectric effect
Consider a particle of mass $m$ and charge $q$ constrained to move along the $x$-axis under the influence of the potential
\[ V(x) = - \alpha \delta(x) \ \ , \ \ \ \ \alpha \gt 0 \]
- Show that there is a single bound state of (negative) energy
\[ E_0 = - \frac{m\alpha^2}{2\hbar^2} \]
with corresponding eigenfunction
\[ \phi_0 (x) = \frac{\sqrt{m\alpha}}{\hbar}\ e^{-m\alpha |x|/\hbar^2} \]
and that for each positive energy
\[ E = \frac{\hbar^2k^2}{2m} \gt 0 \]
there are two stationary states corresponding to a particle incident from the left and right, respectively.
Show that the eigenfunction for a particle coming from the left is
\[ \chi_k (x) = \left\{ \begin{array}{lll} \frac{1}{\sqrt{2\pi}}
\left[ e^{ikx} - \frac{1}{1+\frac{i\hbar^2 k}{m\alpha}} e^{-ikx} \right]
& , & x\lt 0 \\
\frac{1}{\sqrt{2\pi}} \frac{\frac{i\hbar^2 k}{m\alpha}}{1 + \frac{i\hbar^2 k}{m\alpha}} e^{ikx} & , & x\gt 0 \end{array} \right. \]
where the normalization is chosen so that (show!)
\[ \langle \chi_k | \chi_{k'} \rangle = \delta (k-k') \]
HINT: insert a factor $e^{-\epsilon |x|}$ where $\epsilon\gt 0$ and let $\epsilon\to 0$ at the end of the calculation. Use (with proof)
\[ \lim_{\epsilon\to 0^+} \frac{\epsilon}{\epsilon^2 + q^2} = \pi \delta (q) \]
- Calculate the matrix element of the position operator
\[ \langle \chi_k | X | \phi_0 \rangle \]
- The particle interacts with an oscillating electric field of angular frequency $\omega$ where $\hbar\omega \gt -E_0$. The interaction Hamiltonian is
\[ W(t) = -q\mathcal{E} x \sin (\omega t) \]
If the particle is initially in the bound state $|\phi_0\rangle$,
what is the transition probability $w$ per unit time to a positive energy state of energy $E\gt 0\ $? (photoelectric or photoionization effect)
How does $w$ vary with $\omega$ and $\mathcal{E}\ $?
Problem 4.4
Random perturbation - simple relaxation model
A physical system is described by the Hamiltonian $H_0$. At time $t=0$ it is in the eigenstate $|i\rangle$ ($H_0|i\rangle = E_i |i\rangle$).
The system is subject to a perturbation $W(t)$. Let $\mathcal{P}_{if} (t)$ be the probability of finding the system in the eigenstate $|f\rangle$ at time $t$ ($H_0 |f\rangle = E_f |f\rangle$). The transition probability per unit time is
\[ w_{if} (t) = \frac{d \mathcal{P}_{if} (t)}{dt} \]
- Using first-order perturbation theory show that
\[ w_{if} (t) = \frac{1}{\hbar^2} \int_0^t d\tau\ e^{i\omega_{fi}\tau} W_{fi} (t) W_{fi}^* (t-\tau) + \mathrm{c.c.} \]
where $\hbar\omega_{fi} = E_f - E_i$.
- Consider a very large number $N$ of systems which are identical but do not interact with each other.
Each system has a different microscopic environment and consequently sees a different perturbation $W^{(k)} (t)$ ($k=1,2,\dots ,N$).
It is impossible to know with certainty each perturbation $W^{(k)} (t)$ and can only specify statistical averages,
\[ \overline{W_{fi} (t)} = \frac{1}{N} \sum_{k=1}^N W_{fi}^{(k)} (t) \ \ , \ \ \ \
\overline{W_{fi} (t) W_{fi}^* (t-\tau)} = \frac{1}{N} \sum_{k=1}^N W_{fi}^{(k)} (t) W_{fi}^{(k)*} (t-\tau) \]
This perturbation is said to be random.
A random perturbation is called stationary if the statistical averages are time independent.
Then the unperturbed Hamiltonian $H_0$ is redefined so that all
\[ \overline{W_{fi}} = 0 \]
We define the correlation function of the perturbation by
\[ g_{fi} (\tau) = \overline{W_{fi} (t) W_{fi}^* (t-\tau)} \]
It generally vanishes for $\tau \gg \tau_c$ where $\tau_c$ is the correlation time of the perturbation.
Thus the perturbation has a "memory" that extends into the past over a time interval of order $\tau_c$.
- Suppose that the $N$ systems are all in the state $|i\rangle$ at $t=0$ and are subject to a stationary random perturbation of correlation function $g_{fi} (\tau)$ and correlation time $\tau_c$.
Calculate the proportion $\pi_{if} (t)$ of these systems that go into the state $|f\rangle$ per unit time.
Show that after a certain time ($t\gg t_1$, with $t_1$ to be determined), $\pi_{if} (t)$ is independent of time.
- Let
\[ g_{fi} (\tau) = |v_{fi}|^2 e^{-\tau /\tau_c} \]
Find the constant $\pi_{if}$ for $t\gg t_1$ and plot it as a function of $\omega_{fi}$ for fixed $\tau_c$.
- The above results are valid only for $t\ll t_2$, since they are derived using perturbation theory. Estimate $t_2$.
Since $t_1 \ll t_2$ (why?), find the condition for having a transition probability per unit time which is independent of $t$ for $g_{fi}$ given in part ii.
- Application to a simple system.
Suppose that each of the $N$ systems is a spin 1/2 particle of gyromagnetic ratio $\gamma$ placed in a uniform field in the $z$-direction, $\vec B = B_0\hat z$ and set $\omega_0 = -\gamma B_0$.
These spinors are enclosed in a spherical cell of radius $R$ bouncing constantly back and forth between the walls.
Let $t_v$ be the flight time (mean time between two collisions of a spinor with the spherical wall).
Between collisions, the spinor sees only the field $\vec B$. During a collision with the wall, a spinor remains absorbed on the surface for a mean time $\tau_a \ll \tau_v$
during which it sees a microscopic magnetic field $\vec b$ in addition to $\vec B$ due to the paramagnetic impurities in the wall.
The direction of $\vec b$ varies randomly from one collision to another. Let $b_0$ be the mean value of $|\vec b|$.
- What is the correlation time of the perturbation seen by the spinors?
Assuming a correlation function of the form given in part b.ii., show that
\[ \overline{b_i (t) b_j (t-\tau)} = \left\{ \begin{array}{ccc} \frac{1}{3} b_0^2 \frac{\tau_a}{\tau_v} e^{-\tau/\tau_a} & , & i=j \\
0 & , & i\ne j \end{array} \right. \]
where $i,j=x,y,z$.
- Let $\vec M\ $ be the macroscopic magnetization of the $N$ spinors.
Show that, under the effect of the collisions with the wall, $M_z$ (the component of $\vec M\ $ along $\vec B$) relaxes with a time constant (longitudinal relaxation time) $T_1$,
\[ \frac{dM_z}{dt} = - \frac{M_z}{T_1} \]
and find $T_1$.
- Show that by studying the variation of $T_1$ with $B_0$ one can determine the absorption time $\tau_a$ experimentally.
- Suppose we are given several cells of different radii $R$ constructed from the same material.
By measuring $T_1$, how can we determine experimentally the mean amplitude $b_0$ of the microscopic field at the wall?
Problem 4.5
Absorption of radiation by a Hydrogen atom
Consider two particles of masses $m_1$ and $m_2$ and opposite charges $q_1$ and $q_2$ (electron and proton in the case of a Hydrogen atom).
Let $\vec r_i$, $\vec p_i$ ($i=1,2$) be their position vectors and momenta.
Define
\[ \vec r = \vec r_1 - \vec r_2 \ \ , \ \ \ \ \vec r_G = \frac{m_1 \vec r_1 + m_2 \vec r_2}{m_1+m_2} \ \ , \ \ \ \ \vec p = \frac{m_2\vec p_1 - m_1 \vec p_2}{m_1+m_2} \ \ , \ \ \ \ \vec p_G = \vec p_1 + \vec p_2 \]
and also let
\[ M = m_1 + m_2 \ \ , \ \ \ \ m = \frac{m_1m_2}{m_1+m_2} \ \ , \ \ \ \ q_1 = -q_2 = q \]
- Show that the Hamiltonian can be split as
\[ H_0 = H_{ext} (\vec r_G, \vec p_G) + H_{int} (\vec r , \vec p) \]
Show that the eigenstates of $H_{ext}$ are $|\vec K\rangle$ with corresponding eigenvalue $\frac{\hbar^2 K^2}{2M}$,
\[ H_{ext} |\vec K\rangle = \frac{\hbar^2 K^2}{2M} |\vec K\rangle \]
What are the bound states of $H_{int}$?
- Consider two eigenstates of $H_{int}$, $|a\rangle$ and $|b\rangle$ of energies $E_a$ and $E_b$, respectively. Suppose $E_b\gt E_a$ and let
\[ E_b - E_a = \hbar\omega_0 \]
What energy must be supplied to the atom to move it from the state $|\vec K ; a\rangle$ (i.e., the atom in the state $|a\rangle$ with a total momentum $\hbar\vec K$) to the state $|\vec K' ; b\rangle$?
- The atom interacts with a plane electromagnetic wave of wave vector $\vec k$ and angular frequency $\omega = ck$. The vector potential is
\[ \vec A ( \vec r, t) = A_0 \vec e \ e^{i (\vec k\cdot \vec r -\omega t)} + \mathrm{c.c.} \]
where $\vec e \perp \vec k$ is the polarization vector.
The principal term of the interaction Hamiltonian between the electromagnetic wave and the atom is
\[ W(t) = - \frac{q_1}{m_1} \vec p_1 \cdot \vec A - \frac{q_2}{m_2} \vec p_2 \cdot \vec A \]
Express $W(t)$ in terms of $\vec r$, $\vec p$, $\vec r_G$, $\vec p_G$, $m$, $M$ and $q$.
Show that in the electric dipole approximation,
\[ \vec k \cdot \vec r \ll 1 \]
we have
\[ W(t) = W e^{-i\omega t} + \mathrm{c.c.} \ \ , \ \ \ \ W = - \frac{qA_0}{m} \vec e \cdot \vec p \ e^{i\vec k \cdot \vec r_G} \]
- Find a relation between $\vec K$, $\vec K'$ and $\vec k$ in order to have a non-vanishing matrix element
\[ \langle \vec K' ; b|W (t) |\vec K ; a \rangle \]
Interpret this relation in terms of momentum conservation during the absorption of an incident photon by the atom.
- Show that if an atom in the state $|\vec K ; a\rangle$ is placed in the above electromagnetic wave, resonance occurs when the energy $\hbar\omega$ of the photons differs from the energy $\hbar\omega_0$ of the atomic transition $|a\rangle \to |b\rangle$ by a quantity $\delta$ to be determined (since $\delta$ is small, you may replace $\omega$ by $\omega_0$ in the expression for $\delta$).
Write $\delta$ as
\[ \delta = \delta_1 + \delta_2 \]
where $\delta_1$ depends on $\vec K$ and the angle between $\vec K$ and $\vec k$ (Doppler effect) and $\delta_2$ is independent of $\vec K$.
Show that $\delta_2$ is the recoil kinetic energy of an atom initially at rest that absorbs a resonant photon.
Show that $\delta_2 \ll \delta_1$ when $\hbar\omega_0 \simeq 10$ eV (the domain of atomic physics). Use $M\simeq 10^9$ eV (mass of proton) and for $K$ a value that corresponds to a thermal velocity at temperature $T = 300$ K.
Would $\delta_2$ be negligible compared to $\delta_1$ if $\hbar\omega_0 \simeq 10^5$ eV (the domain of nuclear physics)?
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Department of Physics and Astronomy