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The HET Group
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HOMEWORK 5
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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: Fri., April 29, 2011
Problem 5.1
Let $h_0$ be the Hamiltonian of a particle. Assume that $h_0$ acts only on the orbital variables and has three equidistant levels of energies $0$, $\hbar\omega_0$ and $2\hbar\omega_0$ which are non-degenerate in the orbital state space (in the total space, the degeneracy is $2s+1$ for a prticle of spin $s$).
- Consider a system of three independent electrons whose Hamiltonian is
\[ H = h_0(1) + h_0(2) + h_0(3) \]
Find the energy levels of $H$ and their degrees of degeneracy.
- How do your answers in part a. change if the electrons are replaced by three identical bosons of spin 0?
Problem 5.2
Probability densities for two identical particles
Let $\phi (\vec r)$ and $\chi(\vec r)$ be two normalized orthogonal wavefunctions and $|\pm\rangle$ the two eigenstates of the $z$-component of the spin $S_z$ of an electron.
- Consider a system of two electrons, one in the state $|\phi, +\rangle$, the other in the state $|\chi , -\rangle$.
Let $\rho_I (\vec r)$ be the one-particle probability density ($\rho_I (\vec r) d^3 r$ is the probability of finding an electron in a volume $d^3 r$ centered at point $\vec r$)
and $\rho_{II} (\vec r, \vec r')$ be the two-particle probability density ($\rho_{II} (\vec r, \vec r') d^3 r d^3 r'$ is the probability of finding one electron in a volume $d^3 r$ centered at point $\vec r$ and the other in a volume $d^3 r'$ centered at point $\vec r'$).
Show that
\begin{eqnarray} \rho_{II} (\vec r , \vec r' ) &=& |\phi (\vec r)\chi (\vec r')|^2
+ |\phi(\vec r') \chi (\vec r)|^2
\\
\rho_I (\vec r) &=& |\phi (\vec r)|^2 + |\chi (\vec r)|^2
\end{eqnarray}
Show that these expressions remain valid even if $\phi$ and $\chi$ are not orthogonal.
Calculate the integrals over all space of $\rho_I (\vec r)$ and $\rho_{II} (\vec r, \vec r')$. Are they equal to 1?
Compare these results with those obtained for a system of two distinguishable particles of spin 1/2, one in the state $|\phi ,+\rangle$, the other in the state $|\chi ,-\rangle$ given that the device which measures their positions is unable to distinguish between the two particles.
- Now assume that one electron is in the state $|\phi, +\rangle$ and the other in the state $|\chi, +\rangle$. Show that
\begin{eqnarray} \rho_{II} (\vec r , \vec r' ) &=& |\phi (\vec r)\chi (\vec r')
- \phi(\vec r') \chi (\vec r)|^2
\\
\rho_I (\vec r) &=& |\phi (\vec r)|^2 + |\chi (\vec r)|^2
\end{eqnarray}
Calculate the integrals over all space of $\rho_I (\vec r)$ and $\rho_{II} (\vec r, \vec r')$.
How do these expressions change if $\phi$ and $\chi$ are not orthogonal?
- How do your answers in parts a. and b. change if the electrons are replaced by two identical bosons of spin $s\ge 1$. Replace $|+\rangle \to |m_s\rangle$, $|-\rangle \to |m_s'\rangle$ with $m_s \ne m_s'$.
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Department of Physics and Astronomy