For the one-dimensional (1D) harmonic oscillator
(HO) the potential energy function has the form \(V(x) = k
x^2/2\), where k is a parameter indicating the strength of the potential in units of eV per nm^2. This potential will tend to confine an electron to the vicinity of its minimum value, which occurs at x=0. In this HW we will examine some of the low-energy phenomenology associated with a single electron in a 1D HO.
For this potential the energy eigenstates of the Schrodinger
wave equation take the form of Hermite polynomials, which I believe you
have recently studied in physics 116b or c. The ground state wave-function (the lowest
energy eigenstate) can be written as:
\(\Psi_1 (x,t) = \frac{1}{\pi^{1/4}\sqrt{a}} e^{-x^2/(2a^2)} \: e^{-i\omega_1 t} \)
where \(a=(\hbar^2/mk)^{1/4}\).
The first and 2nd excited states can be expressed as:
\(\Psi_2 (x,t) = \frac{\sqrt{2}}{\pi^{1/4}\sqrt{a}} (x/a) e^{-x^2/(2a^2)} \: e^{-i\omega_2 t} \)
and
\(\Psi_3 (x,t) = \frac{\sqrt{2}}{\pi^{1/4}\sqrt{a}} (1/2-x^2/a^2) e^{-x^2/(2a^2)} \: e^{-i\omega_3 t} \)
A quantum length scale, \(a=(\hbar^2/mk)^{1/4}\), emerges from the study of an electron in a potential, \(V(x) = k
x^2/2\), that has no inherent length scale. This emergent quantum length scale is important and foreshadows a similar feature for the hydrogen atom. In a few weeks we will ask the question: where does the size of the hydrogen atom come from? What physics determines that size?
1. Consider a single electron in a 1D HO potential with \(k=1 \:eV/nm^2\).
a) What is the value of the length scale a for this value of k?
b) Plot the ground state wave function as a function of x at t=0. What is its value at x=0 nm?
c) Plot the 1st excited state wave function as a function of x at t=0. What is its value at x=a?
d) Plot the 2nd excited state wave function as a function of x at t=0. What is its value at x=0 nm? What is its value at x=2a?
2. a) Calculate the expectation value of the kinetic energy for an electron in the ground state of a 1D HO potential.
b) What is its value in eV for \(k=1\: eV/nm^2\) ?
c) What is its value in eV for \(k=16\: eV/nm^2\) ?
3. a) Calculate the expectation value of the potential energy for an electron in the ground state of a 1D HO potential.
b) What is its value in eV for \(k=1\: eV/nm^2\) ?
c) What is its value in eV for \(k=16\: eV/nm^2\) ?
4. a) Calculate the expectation value of x for an electron in the ground state of a 1D HO.
b) What is its value in nm for \(k=1\: eV/nm^2\) ?
5. a) Calculate the expectation value of \(x^2\) for an electron in the ground state of a 1D HO.
b) What is its value for \(k=1\: eV/nm^2\) ?
c) What is its value for \(k=16\: eV/nm^2\) ?
d) What is the value of the square root of the expectation value of \(x^2\) for these two cases?
extra credit:
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO.
b) Graph x vs time for the case \(k=1\: eV/nm^2\). What is its value at t=0? What is the period of the oscillation in femtoseconds?
late add 2nd extra credit problem. (similar to 6, yet different)
7. a) Calculate the expectation value of x as a function of time for
an electron in a state that is a (normalized) equal mixture of the
ground state and 2nd excited state of a 1D HO. Is this 0?
b) Calculate the expectation value of \(x^2\) as a function of time for
an electron in a state that is a (normalized) equal mixture of the
ground state and 2nd excited state of a 1D HO.
c) Graph the square root of the expectation value of \(x^2\) as a function of time
for the case \(k=1\: eV/nm^2\). What is its value at t=0? What is the
period of the oscillation in femtoseconds?
SOlutions link:
https://drive.google.com/file/d/1dhfPxe31eqz3sF7TbywQgd3xWSUWwmfI/view?usp=sharing
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Midterm 2 solutions
Here are solutions to midterm 2. I see that I did not write it in the solutions, but it is helpful for sp2 type integrals to recall that \(\...
This is a small thing, but should the \( $\omega_1$ \) inside of \( $\Psi_3$ \) be a \( $\omega_3$ \)
ReplyDeleteYes! and that is not a small thing. Thank you very much for catching that.
DeletePS. Use slash paren to get in and out of math mode.
\(
and then at the end \)
DeleteWhen calculating the integral for the expectation value of x the cross terms have exponential terms with imaginary exponents. Would it be correct to expand these exponential terms using Euler's formula into sines and cosines and then only use the real part of the result? I'm not sure how to visualize an expected value for the position of a particle with a time-dependence in the imaginary plane.
ReplyDeleteI think your intuition is really good, but mathematically you have to keep both parts. I think in quantum calculations you can never just throw out the imaginary parts. They are just as "real" as the real parts. But maybe they will cancel or something? fingers crossed.
DeleteHi Dr. Schlesinger, for problem 6 part a, should Psi(x,t) used in the determination be Psi(x,t) = Psi1(x,t) + Psi2(x,t)? In class, when you mixed states, to normalize, you included a 1/Sqrt(2) in front of Psi1(x,t) and Psi2(x,t). Should we do the same here to normalize? I believe so since you state the function is an equal mixture of the ground state and the first excited state. Thanks.
ReplyDeleteYes. you need that extra normalization factor.
DeleteThe \(\frac{1}{\sqrt{2}}\) comes from the fact that the mixed state must still be normalized to 1. \(\frac{1}{\sqrt{2}}\) is used when each state is equally probable.
DeleteThis comment has been removed by the author.
ReplyDeleteAnother question re problem 6 part b. To plot, is w1 related to w2, or is w2 some multiple of w1? Not sure how to plot with two values for w.
ReplyDeleteExcellent question!!! They are related. \(\omega_2 = 3 \omega_1\)
Deleteand \(\omega_1 = (1/2) \sqrt{k/m}\)
Delete