HW 6

Please turn this HW in to Michael Saccone's mailbox in the physics department mailroom by 6 PM on Monday,  May 13. 
 
"What does the wave-function mean?". The knowledge you gain from problems 1 & 2 will help us address that essential question soon.
Problem 3 is relevant to your understanding of what an energy eigenstate is.
Problems 5-10 provide an introduction to quantum mechanics in 2 dimensions, motion in 2D, and the meaning and importance of degeneracy.


1.  Suppose \(V(x) = .076\, x^2/2\)  (and thus \(a\approx1\:nm)\) for \(t<0\) and that \(V(x) = -\alpha \:  \delta(x)\) for \(t \geq 0\). That is, the potential changes abruptly from a HO to a delta function at t=0.
a) Consider an electron in the GS of the HO potential for t<0. Show that if \(\alpha = .076\: eV*nm\) then the probability of the electron being trapped (localized, confined) in the GS of the delta function after t=0 is about \(.98^2\), that is, about 96%. Sketch the ground state wave-functions for each of these two potentials.
b) Now suppose that instead the delta function has a strength of \(\alpha = 1.0 \: eV*nm\) for t > 0. Sketch the ground state wave-function for each of these potentials and discuss their similarities and differences. Which seems more localized? What is the probability of the electron being trapped in the GS of the delta function after t=0 in this case?
c) extra credit: For \(\alpha=1\), what is the error if you take the the \(e^{-x^2/2a^2}\) out of the integral and use its value (that is, the value of the Gaussian) at the position of the peak of the ground state of the delta-function potential instead (in the case, that is x=0)? Using that approximation show that the probability of trapping the electron which is originally in the ground state of the HO potential is proportional to \(\psi^{*ho}_1(x_o)\psi_1^{ho}(x_o)\), where \(x_o\) is the position of the peak of the delta function. This provides some insight into why \(\psi^*\psi\) is called the probability density.

2.  Consider an electron in the ground state of a finite square well potential, \(V(x) = -0.3 eV,\: -1 < x<1\) nm for \(t \leq 0\). Suppose the potential suddenly changes into \(V(x) = -0.6 \:eV*nm \:  \delta(x)\) for \(t \geq 0\). That is, the potential changes abruptly from a finite square well which is 2 nm wide and -0.3 eV deep, to an extremely narrow finite square well with the same "area" in eV*nm which we choose to approximate as a delta function.
a) Sketch the ground state wave-function for each of these potentials and discuss their similarities and differences. Which seems more localized? Which is larger at x=0?
b) Calculate the approximate* probability of the electron being trapped (localized, confined) in the GS of the delta function(DF) after t=0.  *From the relatively localized nature of the DF GS, you can probably see that contributions to the integral from outside the well will be negligible. I would strongly encourage you to make that approximation or an even more severe approximation. This will save you time and it will be plenty accurate. You can even take the cos function out of the integral and just use its value at the location of the delta function to make things even simpler.
c) extra credit: What energy of photon would you need to use in order to excite this electron into an unbound state for t<0.  What energy of photon would you need to use in order to excite this electron into an unbound state for t>0.

3. a) For an infinite square well extending from -1nm to 1nm, consider the state \(\psi(x) = 1.265 cos^3(\pi  x/2)\).  Does this wave-function satisfy the boundary conditions for this well? Is it a valid state for an electron? Is it an energy-eigenstate wave-function?
b) extra credit: What energy of photon might you want to use in order to excite this electron into the 3rd excited state of this well? (That is, \(\psi_4(x)=sin(4\pi x/2)\) which has an energy \(E_4=16\hbar^2 \pi^2/(2mL^2)\).

A two-dimensional harmonic oscillator (2DHO) will be our first system involving more that 1 spatial dimension. It will be helpful to review aspects of the 1DHO. That is the reason for problem 4.
4. Consider a potential of the form \(V(x) = kx^2/2\).  Show that a function of the form \(e^{-x^2/(2a^2)}\) satisfies the TISE if and only if \(a^4=\hbar^2/(m k)\) and \(E = \frac{\hbar}{2} \sqrt{\frac{k}{m}}\)

5. 2DHO.  In two-dimensions the TISE can be written as,
\( - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x,y) - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial y^2} \psi (x,y) + V(x,y) \psi(x,y) = E \psi(x) \)
with \(V(x,y) = \frac{k}{2}(x^2 + y^2)\) .
a) Show that one can satisfy the TISE with a function of the form \(e^{-x^2/2a^2}e^{-y^2/2a^2}\). What energy and value of the length scale parameter "a" are required?
b) Normalize this spatial wave function, and then write down the normalized time dependent wave-function.
c) Use contour plots to illustrate the nature of this wave-function.
d) Is this an energy eigenstate wave-function?

6. a) Show that one can satisfy the TISE for a 2DHO with a function of the form \(xe^{-x^2/2a^2}e^{-y^2/2a^2}\). What energy and value of the length scale parameter "a" are required?
b) Normalize this spatial wave function.
c) Use contour plots to illustrate the nature of this wave-function.
d) Is this an energy eigenstate wave-function?

7. Consider an electron in a state of the form \((\frac{a}{\sqrt{2}} + x)e^{-x^2/2a^2}e^{-y^2/2a^2}\).
a) Normalize this state. Use a contour plot to illustrate the nature of this wave-function at t=0.
b) Is this an energy eigenstate? Why or why not?
c) Write the full time-dependent wave-function associated with this spatial state.
d) Make contour plots of \(\Psi^*(x,y,t)\Psi(x,y,t)\) at t=0 and and \(t=\pi \sqrt{m/k}\).  What do these contour plots suggest about the nature of the electron motion for an electron in this state?
e) extra credit: Calculate and graph \(\langle x \rangle\) and \(\langle y \rangle\). Describe in words the motion you see modeled here.

8. Consider a state of the form \((x+y)e^{-x^2/2a^2}e^{-y^2/2a^2}\).
a) Normalize this state.  Is this an energy eigenstate wave-function? Why or why not?
b) Use a contour plot to illustrate the nature of this state at t=0.
c) Write the full time dependent state for this spatial wave-function.
d) extra credit: Calculate \(\langle x \rangle\) and \(\langle y \rangle\) for an electron in this state.

9. Consider an electron in the state \(\frac{1}{a\sqrt{2 \pi}}(1 + x/a + y/a)e^{-x^2/2a^2}e^{-y^2/2a^2}\).
a) Is this state normalized? Is it an energy eigenstate?
b) Make contour plots of \(\Psi^*(x,y,t)\Psi(x,y,t)\) at t=0 and \(t=\pi \sqrt{m/k}\).  What do these contour plots suggest about the nature of the electron motion for an electron in this state?
c) extra credit: Calculate and graph \(\langle x \rangle\) and \(\langle y \rangle\). Describe in words the motion you see modeled here.

10. Consider an electron in the state \(\Psi(x,y) = \frac{1}{a\sqrt{2 \pi}}(1 + x/a + i y/a)e^{-x^2/2a^2}e^{-y^2/2a^2}\).
a) Is this state normalized? Is it an energy eigenstate?
b) Make contour plots of \(\Psi^*(x,y,t)\Psi(x,y,t)\) at t=0 and and \(t=\pi \sqrt{m/k}\).  What do these contour plots suggest about the nature of the electron motion for an electron in this state?
c) Make contour plots of \(\Psi^*(x,y,t)\Psi(x,y,t)\) at \(t=\pi \sqrt{m/k}/2\) and \(t=3 \pi \sqrt{m/k}/2\) .
d) extra credit: Calculate and graph \(\langle x \rangle\) and \(\langle y \rangle\). Describe in words the motion you see modeled here.