# 82: Quantum Mechanical Pressure

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Quantum mechanics is based on the concept of wave-particle duality, which for massive particles is expressed simply and succinctly by the de Broglie wave equation.

\[\lambda = \dfrac{h}{mv}=\dfrac{h}{p} \label{1}\]

On the left side is the wave property, \(\lambda\), and on the right the particle property, momentum. These incompatible concepts are united in a reciprocal relationship mediated by the ubiquitous Planck’s constant. Using de Broglie’s equation in the classical expression for kinetic energy, \(T\) converts it to its quantum mechanical equivalent.

\[T = \dfrac{p^2}{2m}=\dfrac{h^2}{2m\lambda^2} \label{2}\]

Because objects with wave-like properties are subject to interference phenomena, quantum effects emerge when they are confined by some restricting potential energy function. For example, to avoid self-interference, a particle in an infinite one-dimensional square-well potential (PIB, particle in a box) of width \(a\) must form standing waves. The required restriction on the allowed wavelengths,

\[\lambda=\dfrac{2a}{n} \;\;\; n=1,2,... \label{3}\]

quantizes the kinetic energy.

\[ T(n)=\dfrac{n^2h^2}{8ma^2} \label{4}\]

In addition to providing a simple explanation for the origin of energy quantization, the PIB model shows that reducing the size of the box **increases **the kinetic energy dramatically. This “repulsive” character of quantum mechanical kinetic energy is the ultimate basis for the stability of matter. It also provides, as we see now, a quantum interpretation for gas pressure. To show this we will consider a particle in the ground state of a three-dimensional box (\(n_x=n_y=n_z=1\)) of width \(a\) and volume \(a^3\). Its kinetic energy is,

\[ T=\dfrac{3h^2}{8ma^2}=\dfrac{3h^2}{8mV^{2/3}}=\dfrac{A}{V^{2/3}} \label{5}\]

According to thermodynamics, pressure is the negative of the derivative of energy with respect to volume.

\[ P = -\dfrac{dT}{dV}=-\dfrac{2}{3}\dfrac{A}{V^{5/3}} \label{6}\]

Using Equation \ref{5} to eliminate \(A\) from Equation \ref{6} yields,

\[ P=\dfrac{2}{3} \dfrac{T}{V} \label{7}\]

This result has the same form as that obtained by the kinetic theory of gases for an individual gas molecule.