Confining a Particle

Thursday, February 17, 2022

Free vs Confined Particles

Free particles are those which have no forces acting on them, meaning they can be located anywhere. Confined particles, on the other hand, are those represented by a wave packet which makes it likely to be found at a particular region of space (the size of which is $\Delta x$ ). The wave packet can be constructed with different sine and cosine waves.

Confined Electrons

Imagine an apparatus with three narrow metal tubes separated by two boundaries: $A$ and $B$ . The distance of the center tube (between $A$ and $B$ ) has length $L$ . The center section is grounded ( $V=0$ ), and the other two sections have some potential $-V_0$ . The gaps $A$ and $B$ are made as small as possible such that the boundaries are effectively instantaneous. This is called a potential energy well.

The potential energy of the electron passed either region is $U_0=qV=\left(-e\right)\left(-V_0\right)=+eV_0$ . Imagine the electron moving in the center region with some kinetic energy $K$ which is less than $U_0$ . This means the electron cannot leave its center potential energy well (classically).

Infinite potential energy well

Now imagine the potential energy $U_0$ is infinite (not a bad approximation when the kinetic energy is much less than the potential energy). This would result in the electron's wave having exactly $0$ amplitude in the forbidden regions on either side, meaning the probability to find the electron there is exactly $0$ .

Using the condition that the wave must be continuous at boundaries $A$ and $B$ , only wave packets of half-wavelengths are possible electron wave packets (ex. $1/2$ , $1$ , $3/2$ , ...). Therefore, only certain wavelengths are allowed, meaning only certain momentum values are allowed, meaning only certain energy levels are allowed. This is called the quantization of energy.

For a region of length $L$ , the possible wavelengths are

$\lambda_n=\frac{2L}{n}$

where $n$ is an integer greater than 1.

Using the de Broglie relationship $\lambda=h/p$ ,

$p_n=n\frac{h}{2L}$

And so the energies allowed (using $E=p^2/2m$ ) are

$E_n=n^2\frac{h^2}{8mL^2}$

These are the quantized values of the energy of the electron. The quantization of energy is a fundamental principle and feature of quantum theory, and is often used in experimental physics.

Using the Uncertainty Principle

Using the infinite potential energy well from before, we know $\Delta x \sim L$ can be used to approximate the uncertainty of the location. Therefore, using the definition of uncertainty ( $\Delta p=\sqrt{\left(p_x^2\right)_{av}-\left(p_{x,av}\right)^2}$ ) and the fact that the average momentum is $0$ (since the electron can be moving in either direction with equal probability), $\Delta p=nh/L$ . This means

$\Delta x\Delta p_x\sim L\frac{nh}{L}=nh$

The value of this uncertainty is greater than $\hbar/2$ and so fits the Heisenberg uncertainty relationship.