Homework #1 with solutions.

For quantum lengths and energies, please express your results in nm and eV.
You will find that these units make sense and help you develop intuition for quantum problems. Note that: \(\hbar\ c = 197 \: eV-nm\) and you can use c to concert mass to eV. \(mc^2 = 0.51 \times 10^6\:  eV\)

Please feel free to email me or to post questions about these questions in the comment section below. Discussion and engagement are encouraged.

note added: You may have noticed that the notation we use in class and on the course blog sometimes differs from that used in the book Griffiths, Chapter 2. On your homework and exams I would like you to use the notation that I provide.

1. Suppose an electron is in the ground state of an infinite square well extending from x=-1 nm to x=1 nm.
a) Graph the ground state wave function as a function of x at t=0.
b) What is the value of the wave function at x=0 nm? What is its value at x=0.5 nm? What is its value at x=1 nm? What is its value at x=1.5 nm? What is its value at x=-0.5 nm?

2. For an electron is in the ground state of an infinite square well extending from x=-1 nm to x=1 nm.
a) Calculate the expectation value of this electron's kinetic energy (K.E.).
b) What is the expectation value of this electron's potential energy.
c) Explain in a sentence or two: why this electron has kinetic energy? (what is the origin and nature of that kinetic energy?)
d) Explain also your result from part b).

3. For an electron is in the ground state of an infinite square well extending from x=-1 nm to x=1 nm.
a) Calculate the expectation value of \(x\).
b) Calculate the expectation value of \(x^2\). The square root of this quantity provides a rough mathematical measure of the extent or "size" of the electron in this state. What is it in nm?

4. Consider an electron is in the ground state of an infinite square.
a) What is the electron's K.E. if the well width is 2 nm? (in eV)
b) What is the electron's K.E. if the well width is 1 nm? (in eV)
c) What is the electron's K.E. if the well width is 0.5 nm? (in eV)
d) What well width would give you a K.E. of 1 eV?

5. (this problem takes more time than the others and has 3x value. It is important for understanding QM.) Consider an electron in an infinite square well extending from x = -L/2 to L/2. Suppose the electron is in the state:
\(\Psi(x,t) = \frac{1}{\sqrt{L}}\:[cos(\pi x/L) \: e^{-i\omega_1 t} + sin(2\pi x/L) \: e^{-i 4 \omega_1 t}\)]
a) Is this wave-function normalized? (If not, please send me an email.) Please check using an appropriate "inner product".  I think you'll have 4 integrals in that calculation. Which ones are zero; which are non-zero?
b) Calculate the expectation value of x at t=0.
c) Graph this wave function as a function of x for t=0. (Not the integrand. The wave function.) Put a dot or line on your graph where your result from b) lies.
d) Calculate the expectation value of x as a function of time.
e) Using \(\omega_1 = \hbar \pi^2/(2 m L^2)\), L = 2 nm, and \(mc^2 = 0.51 \times 10^6\:  eV\) graph the expectation value of x as a function of time. What is its value at t=0? What is a natural time scale for your graph? (I am thinking maybe femto-seconds, but it could be pico? What do you think?)
f) What is the amplitude of your oscillation of x (in nm); what is its period?

6. (this problem takes more time than some others and has 2x value) Consider an electron bound in a potential of the form \(V(x) = k x^2/2\). 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. (If not, please review Hermite polynomials soon!) 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 \(a=(\hbar^2/mk)^{1/4}\).
a) Calculate the expectation value of x for an electron in this state. Does it depend on time?
b) Calculate the expectation value of \(x^2\) for an electron in this state.
c) The square root of the expectation value of \(x^2\) can provide a length scale related to the size (spatial extent) of an electron wave function. What is it in this case? Discuss that nature of its dependence on k and m.
-extra credit.
d) Calculate the expectation value of the kinetic energy of an electron in this state.
e) What value of k gives you a kinetic energy expectation value of 1 eV? What is the value of the length scale, a, for that value of k?
f) Calculate the expectation value of the potential energy of an electron in this state.
extra-extra credit
g) Discuss the relationship between the size, the kinetic energy and the potential energy of an electron in this state.

8. Calculate \(\hbar^2/m\). What are its units in terms of eV and nm?

(problem 7 has been deleted)
7. Consider an electron in an infinite square well extending from x = -1 nm to 1 nm.
a) Hand draw a pretty good graph of the ground state wave-function at t=0.
b) Hand draw a pretty good graph of the 1st excited state wave-function at t=0.
What is the equation for the 1st excited state?
c) Hand draw a pretty good graph of the 2nd excited state wave-function at t=0.
What is the equation for the 2nd excited state? What is the value of this wave-function at t=0 and x=0?
d) Suppose an electron is in a state that is an equal combination of the ground state and 2nd excited state. (These are two even states, both involving cosine functions. Don't get missed up and use the 1st excited state, which is odd, instead.) What is the value of this electron's wave function at x=0 and t=0? What is the value of this electron's wave-function at x=0 and \(t= \pi/\omega_1\).
e) extra credit: For the same state, what is the value of the electron's wave-function at x=0 and \(\pi/(2 \omega_1)\). Why is this a little more challenging than part d? What is different about this result?