Sample Test Problems

1. A nucleus has a radius of 4 Fermis. Use the uncertainty principle to estimate the kinetic energy for a neutron localized inside the nucleus. Do the same for an electron.
   Answer
   
   
   $$\Delta p \Delta x \approx \hbar$$
   
   
   
   
   $$pr\approx\hbar$$
   
   
   Try non-relativistic formula first and verify approximation when we have the energy.
   
   $$E={p^2\over 2m}={\hbar^2\over 2mr^2}={(\hbar c)^2\over 2mc^2r^2}={(197.3 MeV F)^2\over 2(940 MeV)(4 F)^2}\approx 1.3 MeV$$
   
   
   This is much less than 940 MeV so the non-relativistic approximation is very good.

   The electron energy will be higher and its rest mass is only 0.51 MeV so it WILL be relativistic. This makes it easier.
   

   $$pr\approx\hbar$$
   
   
   
   
   $$E=pc={\hbar c\over r}={197.3 MeV F\over 4 F}\approx 50 MeV$$
   
   

2. * Assume that the potential for a neutron near a heavy nucleus is given by $V(r)=0$ for $r>5$ Fermis and $V(r) = -V_0$ for $r<5$ Fermis. Use the uncertainty principle to estimate the minimum value of $V_0$ needed for the neutron to be bound to the nucleus.
3. Use the uncertainty principle to estimate the ground state energy of Hydrogen.
   Answer
   
   $$\begin{eqnarray*} \Delta p \Delta x \approx \hbar \\ pr\approx\hbar \\ E={p^2\over 2m}-{e^2\over r}={p^2\over 2m}-{e^2\over \hbar}p \\ \end{eqnarray*}$$
   
   
   
   
   (We could have replaced $p$ equally well.) Minimize.
   $$\begin{eqnarray*} {dE\over dp}={p\over m}-{e^2\over\hbar}=0 \\ p={me^2\over\hba... ...\hbar c} \\ e^2=\alpha\hbar c \\ E=-{1\over 2}\alpha^2mc^2 \\ \end{eqnarray*}$$
   
   
   
   

4. * Given the following one dimensional probability amplitudes in the position variable x, compute the probability distribution in momentum space. Show that the uncertainty principle is roughly satisfied.

   a

   $\psi (x) = {1\over\sqrt{2a}}$ for $-a<x<a$, otherwise $\psi (x)=0$.

   b

   $\psi (x) = ({\alpha\over \pi})^{1\over 4}\; e^{-\alpha x^2/2}$

   c

   $\psi (x) = \delta (x-x_0)$

5. Use the Heisenberg uncertainty principle to estimate the ground state energy for a particle of mass $m$ in the potential $V(x)={1\over 2} k x^2$.
6. * Find the one dimensional wave function in position space $\psi(x)$ that corresponds to $\phi (p) = \delta (p-p_0)$.
7. * Find the one dimensional wave function in position space $\psi(x)$ that corresponds to $\phi (p) = {1\over\sqrt{2b}}$ for $-b<p<b$, and $\phi (p)=0$ otherwise.
8. * Assume that a particle is localized such that $\psi(x)={1\over \sqrt{a}}$ for $0<x<a$ and that $\psi (x)=0$ elsewhere. What is the probability for the particle to have a momentum between $p$ and $p+dp$?
9. A beam of photons of momentum $p$ is incident upon a slit of width $a$. The resulting diffraction pattern is viewed on screen which is a distance $d$ from the slit. Use the uncertainty principle to estimate the width of the central maximum of the diffraction pattern in terms of the variables given.
10. * The wave-function of a particle in position space is given by $\psi(x)=\delta(x-a)$. Find the wave-function in momentum space. Is the state correctly normalized? Explain why.
11. * A particle is in the state $\psi(x)=Ae^{-\alpha x^2/2}$. What is the probability for the particle to have a momentum between $p$ and $p+dp$?
12. A hydrogen atom has the potential $V(r)={-e^2\over r}$. Use the uncertainty principle to estimate the ground state energy.
13. * Assume that $\phi (p) = {1\over\sqrt{2a}}$ for $\vert p\vert<a$ and $\phi (p)=0$ elsewhere. What is $\psi(x)$? What is the probability to find the particle between $x$ and $x+dx$?
14. The hydrogen atom is made up of a proton and an electron bound together by the Coulomb potential, $V(r)={-e^2\over r}$. It is also possible to make a hydrogen-like atom from a proton and a muon. The force binding the muon to the proton is identical to that for the electron but the muon's mass is 106 MeV/c$^2$. Use the uncertainty principle to estimate the energy and the radius of the ground state of muonic hydrogen.
15. * Given the following one dimensional probability amplitudes in the momentum representation, compute the probability amplitude in the position representation, $\psi(x)$. Show that the uncertainty principle is satisfied.

    (a)

    $\bar{\psi}(p)={1\over\sqrt{2a}}$ for $-a<p<a$, $\bar{\psi}(p)=0$ elsewhere.

    (b)

    $\bar{\psi}(p)=\delta(p-p_0)$

    (c)

    $\bar{\psi}(p)=({\alpha\over\pi})^{1\over 4}e^{-\alpha p^2/2}$

16. * Assume that $\phi (p) = \delta (p-p_0)$. What is $\psi(x)$? What is $<p^2>$? What is $<x^2>$?