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. What is orthonormality?








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








  3. In solving the particle in a box, we started with the general solution

    \begin{displaymath} \vert \phi \rangle = c_1 e^{ikx} + c_2 e^{-ikx} \end{displaymath}

    where $k$ is the wave number and found the particular solution

    \begin{displaymath} \vert \phi_n \rangle = \sqrt {\frac{2}{a}} \sin \left ( \frac{n\pi x }{a} \right ) \end{displaymath}

    where $a$ is the size of the box. What boundary conditions did we impose on the general solution to obtain these eigenfunctions?








  4. In the photoelectric effect, the threshold wavelength is 2756 $\rm\AA$. If light of wavelength 1700$\rm\AA$ is incident on a metal surface, determine the maximum kinetic energy of the photoelectrons.

    A. 4.50 eV D. 2.25 eV
    B. 3.60 eV E. 2.79 eV
    C. 7.29 eV    

  5. What is the value of the commutator $[\hat{H~},x]$ for the quantum mechanical Hamiltonian $\hat{H~} = p^2/2m$?

    A. $\hbar p/im$ D. $2\hbar p/im$
    B. $3\hbar p^2/im$ E. $4\hbar p^2/im$
    C. $\hbar p/2im$    

Problems. Clearly show all work for full credit.


1. (30 pts.)

We developed in class the time-dependent form of the Schroedinger equation shown below.

\begin{displaymath} -{\hbar^2 \over 2\mu} \nabla^2 \Psi(\vec r,t) + V(\vec r)... ...\vec r,t) = i\hbar {\partial \Psi(\vec r,t) \over \partial t} \end{displaymath}

We now want to show that for the one-dimensional case and for potential energy functions that depend only on position (i.e.,$V=V(x)$) that a time-independent form of the Schroedinger equation can be derived as shown below.

\begin{displaymath} -{\hbar^2 \over 2\mu} {\partial^2 \psi(x) \over \partial x^2} + V(x) \psi(x) = E \psi(x) \end{displaymath}

  1. Assume the solution to the time-dependent Schroedinger equation is $\Psi(x,t) = \psi(x)T(t)$. Plug this into the one-dimensional version of the time-dependent Schroedinger equation.

  2. Use the result of part 1.1 to derive the time-independent Schroedinger equation shown above and the equation that $T(t)$ must satisfy.

  3. What is the solution to the time dependent equation for the previous part (1.2)?

3. (45 pts)

Ten million neutrons are in a one-dimensional box with walls at $x=0$ and $x=a$. At $t=0$, the state of each particle is

\begin{displaymath} \psi(x,0) = Ax^2(x-a) \end{displaymath}

where $A=\pm \sqrt{105/a^7}$. The eigenfunctions and eigenvalues are below.

\begin{displaymath} \vert \phi_n \rangle = \sqrt{\frac{2}{a}} \sin \left ( \frac... ... x}{a} \right ) \qquad E_n = n^2 \frac{\hbar^2 \pi^2}{2 m a^2} \end{displaymath}

  1. How many particles have energy $E_7$ at $t=0$?

  2. What is $\langle E \rangle$ at $t=0$?

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} \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... ... dx \quad [ \hat A, \hat B ~ ] = \hat A \hat B - \hat B \hat A \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 ($m_n$) $1.68\times 10^{-27}~kg$    
  $939~MeV/c^2$