Exam 1 Review Questions

This is not comprehensive, but gives a flavor of the sort of questions that students are expected to answer on Exam 1. More questions are forthcoming. No solutions will be given.

Basics

You should be able to answer these questions without thinking (long that is):

1. List the postulates of quantum mechanics and explain them.
2. What are the properties of a wavefunction? Not every function can be a wavefunction.
3. The first Postulate indicates that the wavefunction completely specifies the state of a system? What are the limitations of this "rule"?
4. Give two examples of how a "quantum mechanics" may use a wavefunction?
5. Given a classical observables (position, momentum, kinetic energy, total energy), write down the corresponding quantum operators.
6. What is an expectation value (in words not equations)? How do we determine the expectation value given a wavefunction (in equation, not words)?
7. What does it mean that two wavefunctions are orthogonal to each other? What about a set of wavefunctions is orthonormal?
8. What is a commutator? How is the commutator related to whether two quantities can be observed simultaneously to arbitrary accuracy?
9. What is the difference between the time-dependent and time-independent Schrödinger’s equation? When do you use one vs. the other?
10. Is there a time dependence to the wavefunctions derived for the particle in a box in class? If not, why? If yes, what is it and why didn't we use the time independent Schrödinger’s equation to derive them?

Chapter 1

1. What is the frequency of light attributed to the \(n=3\) line in the Balmer series of hydrogen emission?
2. If 165.7 MJ/mol photon strike atoms of an unknown element and eject electrons with kinetic energies of 25.4 MJ/mol, what is the workfunction of the element?

Chapter 2

1. The kinetic energy operator for a single particle is \[- \dfrac{\hbar^2}{2m} \dfrac{\partial^2}{\partial x^2}\] in one dimension, \[- \dfrac{\hbar^2}{2m} \left ( \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} \right )\] in two dimensions, and \[- \dfrac{\hbar^2}{2m} \left ( \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} + \dfrac{\partial^2}{\partial z^2} \right )\] in three dimensions. If there are two particles, you have to write the sum of the kinetic energy of each particle. For example, the total kinetic energy of a particle of mass m₁ and a particle of mass m₂ in one dimension is \[- \dfrac{\hbar^2}{2m_1} \dfrac{\partial^2}{\partial x_1^2} - \dfrac{\hbar^2}{2m_2} \dfrac{\partial^2}{\partial x_2^2}\]. Write the kinetic energy for a system in 3D containing an electron and a proton.
2. Partial derivatives are fairly simple to calculate. Given \[f(x,y) = x^2 + xy^2 + y^3 + 2\], calculate \[\dfrac{\partial}{\partial x} f(x,y)\] and \[\dfrac{\partial}{\partial y} f(x,y)\]. Now calculate the second partial derivatives \[\dfrac{\partial^2}{\partial x^2} f(x,y)\] and \[\dfrac{\partial^2}{\partial y^2} f(x,y)\].
3. Repeat the proof that the wavefunction can be divided into a time dependent and a time independent part in the case of a Hamiltonian function of two spatial coordinates \(\hat{H}(x, y)\).

Chapter 3

1. Calculate the lowest three energy levels of a particle of mass \(10^{-26}\; Kg\) in a box of length \(L = 10^{-9} m\).
2. Calculate the lowest two energy levels (in eV) of an electron in a 2 Å long one-dimensional box.
3. Plot \(\psi_n(x) \) and \(\psi_n^2 (x)\) for a particle in a box (with infinite height walls) with \(n=1,2,3,4\) for \(0<x<L\)
4. State for which values of x the probability of finding the particle is maximum (for the one dimensional particle in the box) if the system is in state \(n=1\), \(n=2\), or \(n=3\).
5. Show that if \(\psi_n(x) \) is an eigenfunction of the Hamiltonian, \(C \psi_n(x) \) is also an eigenfunction (where \(C\) is any constant).
6. Show that \[ \psi_n(s) = \sqrt{\dfrac{1}{L}} \sin \left(\dfrac{n \pi x}{L} \right) \] is normalized. Is it always possible to normalize a wavefunction by multiplying it by an appropriate constant?
7. If \(\psi_n(x) \) is normalized, then \(\psi_n^*(x) \psi_n(x) \) is the probability density of finding the particle around \(x\) and \[P = \int _a^b \psi_n^*(x) \psi_n(x) dx \] is the probability of finding the particle in the region between \(a<x<b\) (for any one dimensional system). Calculate the probability of finding a particle between \(x=0\) and \(x=L/4\) for a particle in a box in state \(n\).
8. A free particle is a particle without any interactions with potential energy \(V(x)=0\) everywhere (not just within a narrow region of a box)
   1. Write the Hamiltonian for this system.
   2. Show that \(\psi (x) = e^{ikx}\) is an eigenfunction of this Hamiltonian.
   3. Find the eigenvalue corresponding to the eigenfunction \(e^{ikx}\).
   4. Is the energy of the free particle quantized? Why?

Chapter 4

1. Prove that if two operators commute, they have the same eigenfunctions.
2. Show that the commutators \[\left [ \hat{x} , \hat{x}^2 \right ] , \left [ \hat{p}_x , \hat{p}_x^2 \right ] , \left [ \hat{x} , \hat{y} \right ] , \left [ \hat{x} , \hat{p}_y \right ] , \left [ \hat{y} , \hat{p}_x \right ]\] are null.
3. Show that the eigenfunction \(\psi(x) = e^{ikx}\) of the free particle (Exercise 3.8) is also an eigenfunction of the linear momentum operator \( \hat{p}_x\). What would be the result of the measure of the linear momentum for a particle in state \(\psi(x) = e^{ikx}\)?
4. Can energy and linear momentum be known exactly for the free particle? And for the harmonic oscillator?

Chapter 5

1. Calculate the three lowest energy levels of a particle with mass \(m=1.7 \times 10^{-24}\; kg\) in a one dimensional harmonic potential \(V(x)=½kx^²\) with a spring constant (\(k\)) of 100 N/m.
2. The vibration of a diatomic molecule AB can be approximated as the motion of a harmonic oscillator with potential \(V(x)=½kx^2\). \(x\) in this case represents the displacement from equilibrium distance. Instead of the mass, you have to use the reduced mass \(\mu\) related to the mass of A and the mass of B. Calculate the vibrational energy \(\hbar \omega\) of the \(H_2\) molecule for which \(k=510\; N/m\).
3. Draw the probability densities of the first 5 levels of the harmonic oscillator and the first five levels of the particle in the box. Compare them.
4. Consider the harmonic oscillator Schrödinger equation written in the a dimensional coordinate \(y\) as \[\left \{ -\dfrac{1}{2} \dfrac{\partial^2}{\partial y^2} + \dfrac{1}{2} y^2 \right \} \psi (y) = \dfrac{E}{\hbar \omega} \psi (y)\]. Show that \[\psi_0 (y) = \exp \left ( -\dfrac{y^2}{2} \right )\] and \[\psi_1 (y) = y \exp \left ( -\dfrac{y^2}{2} \right )\] are eigenfunctions of \(\hat{H}(y)\) and find the corresponding eigenvalues.
5. Find the values of \(x\) where it is most likely to find a particle in the first excited state of a harmonic oscillator.
6. Is there a zero point energy for the
   • particle in the box?
   • free particle?
7. A macroscopic pendulum has \(\omega=1\, s^{-1}\), \(m=1\; kg\) and total energy \(E=0.1\; J\). What would be its quantum number if described quantum mechanically?
8. The classical probability of finding an oscillator at one particular position is proportional to its inverse velocity. Why?

9. The force constants for typical diatomic molecules are in the range between 400 to 2000 \(N \cdot m^{-1}\).

   Molecule	HCl	HBr	HI	CO	NO
   Force constant, \(k\) (N.m^-1)	480	410	320	1860	1530

   For the diatomic molecules listed above, calculate the following:

   1. angular frequency (\(\text{rad} \cdot s^{-1}\))
   2. natural frequency (Hz)
   3. period (s)
   4. separation between energy levels
   5. wavelength \(\lambda\) of the electromagnetic radiation absorbed in the transition \(v=0 \rightarrow v=1\).