(Quantum review homework)
*For problems 1.4 and 2.1, let's all use the harmonic oscillator wave-functions posted by Nicolas Blanc in the comments below.
For this homework assignment our main focus is on results and relationships. Integrals can be done via wolfram alpha (see video above); there is no point in doing them by hand or showing work. Please comment and ask questions below. (Extra credit for finding errors, ambiguities, etc..)
Section 1: quantum length scales
1.1 Consider an electron in a 1D quantum potential \(U(x) =\frac{1}{2} k x^2\) where k= 100 \(eV/nm^2\). Suppose that the electron is in a state of the form \(\psi(x) = A e^{-x^2/(2a^2)}\).
a) Find A. Calculate the expectation value of \(x\). Calculate the expectation value of \(x^2\).
b) What is the expectation value of the U ?
c) Calculate the expectation value of the kinetic energy, T . What is the relationship between T and the expectation value of \(p^2\) ?
d) Graph the expectation values of T and U as a function of a. What is each one for a = 1 nm? What about for a = 2 nm?
e) What value of a gives you the lowest value for the total energy, T + U ? How does this value of a depend on each member of the "quantum triumvirate" m, k and \(\hbar\)?
1.2 For the results for problem 1.1, calculate the product of the expectation value of \(p^2\) times the expectation value of \(x^2\). On what does this depend? Discuss.
1.3 a) For an electron in the ground state of a hydrogen atom, express the quantum length scale known as the Bohr radius in terms of m, e and \(\hbar\).
b) For an electron in the ground state of a 1D harmonic oscillator, express the
quantum length scale, "a", in terms of m, k and
\(\hbar\).
c) discuss.
1.3e (see below)
1.4 a) For the 3 lowest energy states of a 1D harmonic oscillator, write the normalized energy eigenstate wave-functions. (Please use the HO quantum length scale to simplify both the normalization factor and the exponential.)
[Please post your answer here. That way we can all converge on the same thing.]*
(see comments!)
b) Plot each of these wave functions as a function of x.
c) Using wolfram alpha to save time, calculate the expectation value of \(x^2\) for each of these states. I am thinking that your answer will be in the form of a number times \(a^2\).
d) Take the square root of that to get something with units of length. Plot that (3 points) as a function of the quantum number. What might this tell you about the nature of these states?
