【Easy Quantum Mechanics】Wave-particle-duality

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On this page, we will confirm that light, long thought to be a wave, satisfies Einstein’s equation

\begin{align*}E=\hbar\omega\tag{1}\end{align*}

and de Broglie’s equation

\begin{align*}\boldsymbol p&=\hbar\boldsymbol k,\tag{2}\end{align*}

which express particle and wave properties, and that light has wave-particle-duality.

In addition, by substituting the above two equations (1) and (2) into the complex plane wave equation

\begin{align*}\Psi&=ae^{i\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)},\tag{3}\end{align*}

which has only wave properties, we will derive the complex plane wave equation of light

\begin{align*}\Psi&=ae^{i\left(\frac{\boldsymbol p\cdot\boldsymbol q}{\hbar}-\frac{E}{\hbar}t+\delta\right)}\\&=ae^{i\left(\frac{p_xx}{\hbar}+\frac{p_yy}{\hbar}+\frac{p_zz}{\hbar}-\frac{E}{\hbar}t+\delta\right)},\tag{4}\end{align*}

which has wave-particle-duality.

On the previous page…

On the previous page, we derived the complex plane wave equation

\begin{align*}\Psi&=ae^{i\left(\boldsymbol k\cdot\boldsymbol q-\omega t+\delta\right)}\tag{3}\end{align*}

that satisfied the wave equation

\begin{align*}\frac{\partial^2 \Psi}{\partial t^2}&=u^2\Delta\Psi.\tag{5}\end{align*}

Is light a particle or a wave?

Among the physical phenomena discovered up to the 19th century, light interference, diffraction, polarization, and propagation of electromagnetic waves can be explained by treating light like a wave. On the other hand, reflection and refraction of light can be explained by treating light like a particle. There was a long debate about whether light is a wave or a particle, but in the 19th century, the idea that light is a wave became mainstream.

Discovery of Einstein’s equation

In the 20th century, however, Planck’s law and the photoelectric effect were discovered, which can be explained by treating light like a particle. Assuming that light is a wave, the energy \(E\) of light is proportional to the square of the amplitude \(a\), and since the amplitude \(a\) can take continuous values, the energy \(E\) can also take continuous values. But in these phenomena, light had discrete energy, which was an integer multiple of the angular frequency \(\omega\) multiplied by Dirac’s constant \(\hbar\). In other words, light can be counted as a particle, and the energy \(E\) that a particle of light has is expressed as equation

\begin{align*}E=\hbar\omega.\tag{1}\end{align*}

This equation is called Einstein’s equation.

Derivation of de Broglie’s equation

In the theory of relativity, consider the equation

\begin{align*}E^2=m^2c^4+\boldsymbol p^2c^2,\tag{6}\end{align*}

which expresses the relationship between the energy \(E\) and momentum \(p\) of light.

Here, since the mass of light is zero, equation (6) becomes

\begin{align*}E^2=\boldsymbol p^2c^2.\tag{7}\end{align*}

Substituting Einstein’s equation (1) into equation (7), it becomes

\begin{align*}\hbar^2\omega^2&=\boldsymbol p^2c^2\\\rightarrow \boldsymbol p^2&=\frac{\hbar^2\omega^2}{c^2}=\hbar^2 \boldsymbol k^2\\\rightarrow \boldsymbol p&=\hbar\boldsymbol k,\tag{2}\end{align*}

which is called de Broglie equation.

Wave-particle-duality

From equations (1) and (2), even light, which is thought to be a wave, has energy \(E\) and momentum \(p\), just like a particle. And since their left-hand sides of equations (1) and (2) consist of “the energy \(E\) and momentum \(p\) representing particle properties” and their right-hand sides consist of “the angular frequency \(\omega\) and wavenumber vector \(\boldsymbol k\) representing wave properties,” equations (1) and (2) link particle and wave properties. The property of having both particle and wave properties, such as light, is called wave-particle-duality. As for the interpretation of this phrase, “light has particle-like properties and wave-like properties” is correct, but “light is both a particle and a wave” is incorrect. The phrase “particle-wave duality” merely describes a quantum-scale phenomenon that humans cannot understand, by using the macro-scale words “particle” and “wave” that humans can understand. Therefore, we should look at equation (1) or equation (2) when discussing quantum theory, and it is useless to discuss whether light is a wave or a particle.

Derivation of complex plane waves with duality

The complex plane wave equation (3) has only wave properties, but if we use Einstein’s equation (1) and de Broglie’s equation (2), which have wave-particle-duality, we obtain the complex plane wave equation of light

\begin{align*}\varPsi&=ae^{i\left(\frac{\boldsymbol p\cdot\boldsymbol q}{\hbar}-\frac{E}{\hbar}t+\delta\right)}\\&=ae^{i\left(\frac{p_xx}{\hbar}+\frac{p_yy}{\hbar}+\frac{p_zz}{\hbar}-\frac{E}{\hbar}t+\delta\right)},\tag{4}\end{align*}

which has wave-particle-duality.

What is matter wave?

We now know that light thought to be a wave has both particle and wave properties, but then we would expect matter, which is thought to be a particle, to also have particle and wave properties. In fact, electron beam diffraction and neutron beam diffraction have confirmed that even matter thought to be particles has wave properties. This phenomenon is called a matter wave (or de Broglie wave), and matter is also considered to obey the complex plane wave equation (4).

On the next page…

On the next page, we will derive the eigenvalue equations for coordinates \(q\), momentum \(p\), energy \(E\), and Hamiltonian \(H\) from the complex plane wave equation (4) with wave-particle-duality.

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