Quantum Mechanics - 1st Year Easter Vacation

You need to know some quantum mechanics at the level defined by the problems below.

There are many excellent books on quantum mechanics but the most suitable is Atkins: Quantum Mechanics, Chapters 1-3. You should try to understand these three chapters or their equivalent but, even if you cannot totally understand the concepts and some of the mathematics clearly, make sure that you can do the necessary manipulations.

Problems

Where the question number is enclosed in a button, e.g. , you can obtain help or comments about the question by clicking it.

1. (a) State the Schrödinger equation for a one-dimensional quantum particle of mass m.

(b) What is Born's interpretation of the wavefunction Psi(x), and what conditions must Psi(x) satisfy to be physically acceptable?

(c) A particle is subject to the potential MATHwith $V_{0}>0.\bigskip $

(i) Sketch the potential.

(ii) Show that the wavefunction MATHsatisfies the Schrödinger equation with energy $E<V_{0},$ provided MATH and

MATH The constants A and B need not be determined.

(iii) Explain why Psi(x)$ is not physically acceptable unless $B=A$ exp(MATH

(iv) Provide an argument to show that for $E<V_{0}$ the region $x>L$ is classically forbidden.

(v) Show that the probability $P_{\text{f}}$ that the particle is found in the classically forbidden region is MATH and that the expectation value of the kinetic energy is MATH. You may assume that A has been chosen so that Psi(x) is normalized.

2. (a) Show by direct substitution that the function,

MATH where A, B and k are constants, is a solution of the Schrodinger equation

MATH for a particle of mass m in one dimensional free space for which the potential, $V$, is zero everywhere.

Obtain the energy, $E$, in terms of k and m.

(b) Consider a particle in a one dimensional box of length $L$, where $V=\infty $ for $x\leq 0$ and $x\geq L$ and $V=0$ for $0<x<L.$

(i) What value must the wave-function adopt at $x = 0$ and $x = L?$

(ii) Show that the condition can be satisfied at $x = 0$ by the function $\psi (x)$ in part (a).

(c) Sketch the probability distributions for a particle in the four lowest energy levels in the box described in part (b). How do these compare with the probability distribution of a classical particle in a box?

(d) For a particle of mass m moving in a square two dimensional box of side $L$, the energy levels are given by the expression

MATHwhere $n$ and $q$ are two independent quantum numbers taking on the values 1, 2, 3,... By referring to this example, explain the meaning of degeneracy, as used in quantum mechanics, and deduce which of the six lowest energy levels in this problem exhibit degeneracy.

(a) Sketch (i) the radial wavefunctions and (ii) the radial distribution functions for an electron in the 1s, 2s and 2p orbitals of the hydrogen atom.

(b) The normalised wave-function for an electron in the 1s orbital of the H atom is

MATH where $r$ is the electron-nucleus separation and aSubscript0 is a constant.

(i) Write down an expression for the radial distribution function describing the probability of finding an electron between $r$ and $r + dr.$

(ii) Using your answer to part (i), calculate the most probable electron-nucleus separation in terms of aSubscript0 .

(iii) By calculating the expectation value $<r>,$ determine the mean distance of the electron from the nucleus in terms of aSubscript0 .

MATH

4. (a) What is meant by the Born interpretation of a quantum mechanical wave-function? Describe how the Born interpretation leads to the imposition of boundary conditions for the wave-function of a particle confined to motion on a ring.

(b) The potential energy, $V$, for a harmonic oscillator is given by

MATH where k is the force constant and $x$ is the displacement from equilibrium. Show by direct substitution that the wave-function

MATH where MATH is a solution of the Schrodinger equation for a harmonic oscillator of mass m. Determine the corresponding energy, expressing your answer in terms of the vibrational frequency,

MATH

(c) Determine the position of maximum probability density for this wave-function. How does the result compare (qualitatively) with the time averaged probability of finding a particle at a specific location in a classical harmonic oscillator?

(d) Assuming the harmonic oscillator is a reasonable description of the vibration of a diatomic molecule, explain briefly why

(i) D2subscript has a lower zero point energy than H2subscript.

(ii) N2subscript has a higher vibrational frequency than O2subscript.

MATH

The normalized wave-function for an electron confined to a 1-dimensional box, with infinitely high walls, is

MATH in which $a$ is the length of the box, $x$ the position along the box, and $n $ an integer. The Schrodinger wave equation in one dimension is

MATH

(a) Sketch the appearance of $\psi $ for $n=3$ and for $n=4.$

(b) A simple model of the $\pi $ electrons in a conjugated molecule treats them as being confined to a 1-dimensional box.

(i) Estimate a value for the length of the box from the geometry of butadiene.

(ii) Use the Pauli exclusion principle to give the occupancy of the orbitals.

(iii) Calculate the lowest energy transition that can take place among electrons confined to the conjugated system of butadiene.

(iv) The walls of the box in butadiene are not infinitely high. How would a typical wave-function be modified if walls of infinite height were replaced by walls of finite height?

(v) Calculate the probability of finding an electron in the $n=1$ orbital within 0.1 nm of the left hand wall of the box with infinitely high walls.

MATH