Some Simple Systems

Abstract

The time-independent Schroedinger equation is solved for a number of simple one-dimensional systems. In particular, the Infinite Square Well, the Potential Step, the Square Barrier (tunneling phenomena), the Square Well (bound states) and the Delta Function potential.

Problems and Exercises

3.1

Particles of mass m and energy E are incident from the left on the potential step $V(x)=-\Theta (x)V_0$ with $V_0>0$. Calculate the reflection coefficient.

3.2

Consider a particle of mass m and energy $E>V_0>0$ incident on the potential step $V(x)=\Theta (x)V_0$. Calculate the reflection coefficient for incidence from the right and for incidence from the left and compare the results.

3.3

Consider the asymmetric square well

$$V(x)=\left\{ \begin{array}{cc} V_1\,&{}\,(x<0)\\ \,&{}\,\\ -V_0\,&{}\,(0<x<L)\\ \,&{}\,\\ 0\,&{}\,(x>L) \end{array}\right. $$

with $V_1$ and $V_0$ positive. Investigate the existence of bound states.

3.4

A particle trapped in an infinite square well of width $[-2L,\,2L]$ has initially ($t=0$) a wave function⁷

$$\psi (x,0)\,=\,\left\{ \begin{array}{cc} \frac{1}{\sqrt{L}}\cos (\pi x/2L)\,&{}\,(|x|\le L)\\ \,&{}\,\\ 0\,&{}\,(L\le |x|\le 2L). \end{array}\right. $$

Calculate the probability to find the system in its ground state.

3.5

Verify explicitly the orthogonality of the energy eigenfunctions⁸ of the potential step $V(x)=\Theta (x)V_0$

$$\int _{-\infty }^{+\infty }dx\,\psi _E^*(x)\psi _{E'}(x)\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,(E\ne E')\,.$$

3.6

A particle of mass m is trapped in an infinite square well of width $[-L,\,L]$. The particle is initially ($t=0$) in a state with wave function

$$\psi (x,0)\,=\,\frac{1}{\sqrt{2}}\left( \,\psi _N(x)\,+\,i\,\psi _{N+1}(x)\,\right) \,,$$

where $\psi _n(x)$ is the energy eigenfunction corresponding to energy $E_n$. Find the wave function of the particle at times $t>0$. Calculate the probability to find the particle in the initial state at time $t>0$. What is the minimum time that this probability becomes one? Calculate the probability to find the particle at time $t>0$ in the orthogonal state

$$\psi ^{(\bot )}(x)\,=\,\frac{1}{\sqrt{2}}\left( \,\psi _N(x)\,-i\psi _{N+1}(x)\,\right) \,.$$

3.7

A particle of mass m and energy $E>0$ is incident from the right on a potential step $V(x)=-V_0\Theta (-x)$ with $V_0>0$. What is the value of the ratio $E/V_0$ if the reflection coefficient is 1 / 4?

3.8

A particle of mass m is bound in a potential

$$V(x)\,=\,\left\{ \begin{array}{cc} +\infty \,&{}\,(x\le 0)\\ \,&{}\,\\ -\frac{\hbar ^2g^2}{2m}\delta (x-a)\,&{}\,(x>0). \end{array}\right. $$

Calculate the probability to find the particle in the region $a\le x<\infty$.

3.9

A particle moves in the potential $V(x)=g^2\,|x|$. If the particle is in a state with wave function

$$\psi (x)\,=\,N\,x\,e^{-\alpha \,x^2}\,,$$

where $N,\,\alpha >0$ are known parameters, calculate the probability to find the particle in the ground state $\psi _0(x)$ of the energy. The exact form of $\psi _0(x)$ is not necessary.

3.10

Consider a particle moving in the potential of two attractive delta functions

$$V(x)\,=\,-\frac{\hbar ^2 g^2}{2m}\left( \,\delta (x+a)\,+\,\delta (x-a)\,\right) \,.$$

Investigate the existence of bound states ($E<0$).