Physics 401 Test 1

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Questions (5 pts. apiece) Answer questions 1-3 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 4-5 circle the correct answer.

  1. Cite at least two experimental results that motivated the development of quantum mechanics.








  2. What is the Rayleigh-Jeans Law (in words or equations) and why was it important for the development of quantum mechanics?








  3. The eigenfunctions and eigenvalues of the particle in a box are

    \begin{displaymath} \vert\phi \rangle = \sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a} \qquad E_n = n^2 \frac{\hbar^2 \pi^2}{2ma^2} \end{displaymath}

    for $0<x<a$. The eigenfunctions are zero outside the box. Consider the following sequence of measurements of a particle in a box.

    1. The energy of the particle is measured. A value $E_1$ is obtained.

    2. The value of the position of the particle is made and a value $x_2$ is obtained.

    3. The energy of the particle is measured again.

    What possible values of energy can you obtain in step 3.c? Explain.








  4. What is the definition of a solid angle?








  5. We developed the Schroedinger equation by using a well-known classical physics equation as a template and then adding in the Planck and de Broglie hypotheses. What well-known classical equation did we use?








  6. When Fourier first introduced his ideas for expanding any function, including those with `sharp' edges (recall the initial rectangular wave packet we have seen in the computational laboratory on superposition) the notion was met with some skepticism. How would you convince someone that one can use the eigenfunctions of the infinite square well to describe a rectangular wave?








Problems. Clearly show all work for full credit.


1. (15 pts.)

An electron moves in the $x$ direction with the de Broglie wavelength $\lambda = 5\times 10^{-10}~m$. What is the energy of electron in eV? What is the time-independent wave function of the electron?

2. (25 pts)

What is the uncertainty relationship for $\Delta x \Delta T$ where $T$ refers to the kinetic energy operator $-(\hbar^2/2m)\partial^2 /\partial x^2$?

3. (30 pts)

The eigenfunctions and eigenvalues of the particle in a box are

\begin{displaymath} \vert\phi \rangle = \sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a} \qquad E_n = n^2 \frac{\hbar^2 \pi^2}{2ma^2} \end{displaymath}

for $0<x<a$. The eigenfunctions are zero outside the box. Measurement of the position of the a particle in a one-dimensional box with walls at $x=0$ and $x=a$ finds the value $x=0.7a$. What is the probability for finding a particular energy value in a subsequent measurement? What is the most probable value?

Physics 401 Equations, Conversions, and Constants


\begin{displaymath} R_T(\nu) = {Energy \over time \times area} \quad E = h\nu = ... ...nu \quad I \propto \vert\vec E\vert^2 \quad K_{max} = h\nu - W \end{displaymath}


\begin{displaymath} E = \frac{1}{2} m v^2 + V = \frac{p^2}{2m} + V \qquad \vec p = m \vec v \end{displaymath}


\begin{displaymath} \lambda = {h \over p} \quad p = \hbar k \quad -{\hbar^2 \ove... ...i(x,t) \quad \hat {p }_x = -i\hbar {\partial \over \partial x} \end{displaymath}


\begin{displaymath} \hat{A~}\vert\phi\rangle = a\vert\phi\rangle \quad \langle\h... ...d \Delta A \Delta B \ge \frac{\vert\langle C \rangle \vert}{2} \end{displaymath}


\begin{displaymath} \langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\i... ...nt_{-\infty}^{\infty} \phi_{k'}^* \phi_k  dx = \delta(k - k') \end{displaymath}


\begin{displaymath} \vert\psi\rangle = \sum b_n \vert\phi_n\rangle \rightarrow ... ...angle dk \rightarrow b(k) = \langle\phi(k) \vert \psi \rangle \end{displaymath}


\begin{displaymath} \Delta p \Delta x \ge {\hbar \over 2} \quad (\Delta x)^2 = \... ...ma^2}  e^{-x^2/2\sigma^2}, {\rm then}\ \Delta x = \sigma \end{displaymath}


\begin{displaymath} e^{ix} = \cos x + i\sin x \end{displaymath}

Speed of light ($c$) $2.9979\times 10^8 m/s$ fermi ($fm$) $10^{-15} m$
Boltzmann constant ($k_B$) $1.381\times 10^{-23} J/K$ angstrom ($\rm\AA$) $10^{-10} m$
  $8.62\times 10^{-5} eV/k$ electron-volt ($eV$) $1.6\times 10^{-19} J$
Planck constant ($h$) $6.621 \times 10^{-34} J-s$ MeV $10^6 eV$
  $4.1357\times 10^{-15} eV-s$ GeV $10^9 eV$
Planck constant ($\hbar$) $1.0546\times 10^{-34} J-s$ Electron charge ($e$) $1.6\times 10^{-19} C$
  $6.5821\times 10^{-16} eV-s$ $e^2$ $\hbar c / 137$
Planck constant ($\hbar c $) $197 MeV-fm $ Electron mass ($m_e$) $9.11\times 10^{-31} kg$
  $1970 eV-{\rm\AA}$   $0.511 MeV/c^2$
Proton mass ($m_p$) $1.67\times 10^{-27}kg$ atomic mass unit ($u$) $1.66\times 10^{-27} kg$
  $938 MeV/c^2$   $931.5 MeV/c^2$
Neutron mass ($m_n$) $1.68\times 10^{-27} kg$    
  $939 MeV/c^2$