Particle in a box

Figure 20.5:

A particle of energy $E$ is confined to $0\le x \le L$ by a very high (infinite ) potential as shown in Figure 20.5. This is often called an infinite potential well or a particle in a box. The potential is zero inside the well and it is infinite outside.

We have seen that the wave function decays exponentially inside a region where and the decay occurs more rapidly for higher $V$. In the limit where $V \rightarrow \infty$ the wavefunction vanishes at the boundary. We then have

$$\frac{d^2x}{dx} = - \frac{2m E}{\hbar^2} X ~ \hspace{1cm}~ 0 \le x \le L$$ (20.56)

with the boundary condition that $X(x)$ vanishes at $x=0$ and $x=L$. We have encountered exactly the same situation when studying standing waves and the solution is

$$X(x) = A_n \sin \left( \frac{n \pi x}{L} \right) \hspace{1cm} {\rm n} =\pm,\pm 2,...\,.$$ (20.57)

where there is a different solution corresponding to each integer $n=1,2,3,...$ etc. Substituting this solution in equation (20.56) we have

$$\left( \frac{n \pi}{L} \right)^2 = \frac{2m E_n}{\hbar^2}$$ (20.58)

which gives the energy to be

(20.59)

We find that there is a discrete set of allowed energies $E_1$,$E_2$, $E_3$,... corresponding to different integers $1$, $2$, $3$,.... Inside the potential well there do not exist states with other values of energies.

Figure 20.6:

The state with $n=1$ has the lowest energy

(20.60)

and this is called the ground state. The wavefunction for a particle in this state is

$$\psi_1 (x,t) = A_1e^{- iE_1 t/\hbar} \, \sin \left( \frac{\pi x}{L} \right),$$ (20.61)

This is shown in Figure 20.6. Normalizing this wavefunction $\int \limits^\infty_{-\infty} \psi^*_1 \psi_1 dx = 1$ determines .

The $n=2$ state is the first excited state. It has energy and its wavefunction (shown in Figure 20.6) is

$$\psi_2 (x) = A_2 e^{-i E_2 t/\hbar} \sin \left( \frac{2 \pi x}{L}\,. \right)$$ (20.62)

The wavefunction

$$\psi(x,t)= c_1 \psi_1 (x,t) + c_2 \psi_2 (x,t)$$ (20.63)

is also an allowed state of a particle in the potential well. This is not an energy eigenstate. We will get either $E_1$ or $E_2$ if the particle's energy is measured.

Figure 20.7:

The energy of the higher excited states increases as $E_n=n^2 E_1$. Consider a particle that undergoes a transition from the $n$ state to the $n-1$ state as shown in Figure 20.7.The particle loses energy in such a process. Such a transition may be accompanied by the emission of a photon ( $\gamma$) of frequency $\nu=(E_n-E_{n-1})/h$ which carries away the energy lost by the particle.

It is now possible to fabricate microscopic potential wells using modern semiconductor technology. This can be achieved by doping a very small regions of a semiconductor so that an electron inside the doped region has a lower potential than the rest of the semiconductor. An electron trapped inside this potential well will have discrete energy levels $E_1$, $E_2$, etc. like the ones calculated here. Such a device is called a quantum well and photon's are emitted when electron's jump from a higher to a lower energy level inside the quantum well.