← Quantum Mechanics

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 Δx\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: AA and BB. The distance of the center tube (between AA and BB) has length LL. The center section is grounded (V=0V=0), and the other two sections have some potential V0-V_0. The gaps AA and BB 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 U0=qV=(e)(V0)=+eV0U_0=qV=\left(-e\right)\left(-V_0\right)=+eV_0. Imagine the electron moving in the center region with some kinetic energy KK which is less than U0U_0. This means the electron cannot leave its center potential energy well (classically).

Infinite potential energy well

Now imagine the potential energy U0U_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 00 amplitude in the forbidden regions on either side, meaning the probability to find the electron there is exactly 00.

Using the condition that the wave must be continuous at boundaries AA and BB, only wave packets of half-wavelengths are possible electron wave packets (ex. 1/21/2, 11, 3/23/2, ...). 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 LL, the possible wavelengths are


where nn is an integer greater than 1.

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


And so the energies allowed (using E=p2/2mE=p^2/2m) are


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 ΔxL\Delta x \sim L can be used to approximate the uncertainty of the location. Therefore, using the definition of uncertainty (Δp=(px2)av(px,av)2\Delta p=\sqrt{\left(p_x^2\right)_{av}-\left(p_{x,av}\right)^2}) and the fact that the average momentum is 00 (since the electron can be moving in either direction with equal probability), Δp=nh/L\Delta p=nh/L. This means

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

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