Question

If $\mathrm{\Delta }{x}^{\ast }=(c\mathrm{\Delta }t,\mathrm{\Delta }\mathbf{r})$ is the four-vector join of two events on the worldline of a uniformly moving particle (or photon), prove that the frequency vector of its de Broglie wave is given by $$N^{\mu }=\frac{v}{c\mathrm{\Delta }t}\mathrm{\Delta }{x}^{\mu },$$ whence $v/\mathrm{\Delta }t$ is invariant. Compare with Exercise III $(7)$.

Short answer

The expression for the de Broglie wave's frequency vector is given by $N^{\mu }=\frac{v}{\sqrt{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }-c^2\mathrm{\Delta }{t}^2}}\mathrm{\Delta }{x}^{\mu }$, where $\mathrm{\Delta }{x}^{\mu }$ is the four-vector join. Additionally, the quantity $v/\mathrm{\Delta }t$ is indeed invariant, meaning it remains constant under Lorentz transformation.

Step-by-step solution

Define the four-vector join and the de Broglie wave

The four-vector join $\mathrm{\Delta }{x}^{\mu }$ connects two events in spacetime, given by $\mathrm{\Delta }{x}^{\mu }=(c\mathrm{\Delta }t,\mathrm{\Delta }\mathbf{r})$, where $c$ is the speed of light and $\mathrm{\Delta }t$ and $\mathrm{\Delta }\mathbf{r}$ represent time and spatial separations, respectively. The de Broglie wave is associated with a moving particle and has a frequency $v$, which can be expressed as $v=\lambda f$, where $\lambda$ is the wavelength and $f$ is the frequency.

Use the four-vector join to express de Broglie wavelength

We will start by extracting the relationship between the de Broglie wavelength and the four-vector join. By definition, the de Broglie wavelength $\lambda$ is the length of a de Broglie wave, so $\lambda =|\mathrm{\Delta }\mathbf{r}|$. Therefore, we have: $$\lambda =\sqrt{(\mathrm{\Delta }\mathbf{r}{)}^2}=\sqrt{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }-c^2\mathrm{\Delta }{t}^2}.$$

Find the relationship between frequency and the four-vector join

Next, we will express the de Broglie frequency $f$ in terms of four-vector join. From the de Broglie relation, we have $v=\lambda f$. So the frequency is given by: $$f=\frac{v}{\lambda }.$$ Substituting $\lambda$ in terms of the four-vector join, we get $$f=\frac{v}{\sqrt{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }-c^2\mathrm{\Delta }{t}^2}}.$$

Express the frequency vector $N^{\mu }$

Now we will express the frequency vector $N^{\mu }$ in terms of the four-vector join $\mathrm{\Delta }{x}^{\mu }$, the de Broglie frequency $f$, and the de Broglie wave's velocity $v$. The frequency vector $N^{\mu }$ is given by $$N^{\mu }=f\mathrm{\Delta }{x}^{\mu }.$$ Substituting the expression for $f$ from step 3, we obtain $$N^{\mu }=\frac{v}{\sqrt{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }-c^2\mathrm{\Delta }{t}^2}}\mathrm{\Delta }{x}^{\mu }.$$

Prove that $v/\mathrm{\Delta }t$ is invariant

To prove that $v/\mathrm{\Delta }t$ is invariant, we need to show that it remains constant under Lorentz transformation. First, we will find the expression for $v/\mathrm{\Delta }t$: $$\frac{v}{\mathrm{\Delta }t}=\frac{\lambda f}{\mathrm{\Delta }t}.$$ Using the expressions from step 3 and step 2, this can be rewritten as $$\frac{v}{\mathrm{\Delta }t}=\frac{\sqrt{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }-c^2\mathrm{\Delta }{t}^2}}{\mathrm{\Delta }t}.$$ Now we will show that this expression is invariant. Recall the Lorentz transformation: $$\mathrm{\Delta }{x}^{\prime \mu }={\mathrm{\Lambda }}_{\nu }^{\mu }\mathrm{\Delta }{x}^{\nu },$$ where ${\mathrm{\Lambda }}_{\nu }^{\mu }$ is the Lorentz transformation matrix. The invariance of the interval tells us that $$\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }=\mathrm{\Delta }{x}^{\prime \mu }\mathrm{\Delta }{x}_{\mu }^{\prime }.$$ Thus, we can write $$\frac{v}{\mathrm{\Delta }t}=\frac{\sqrt{\mathrm{\Delta }{x}^{\mu }\mathrm{\Delta }{x}_{\mu }-c^2\mathrm{\Delta }{t}^2}}{\mathrm{\Delta }t}=\frac{\sqrt{\mathrm{\Delta }{x}^{\prime \mu }\mathrm{\Delta }{x}_{\mu }^{\prime }-c^2\mathrm{\Delta }{t}^{\prime 2}}}{\mathrm{\Delta }{t}^{\prime }}.$$ Since the equality holds for both unprimed and primed coordinates, we can conclude that $v/\mathrm{\Delta }t$ is indeed invariant. Comparing this result with Exercise III (7), we can see that both exercises reveal the invariance of certain quantities related to the de Broglie wave in spacetime. This invariance property is essential in the study of relativistic quantum mechanics.

Key concepts

Four-Vector

The concept of a four-vector is a cornerstone in the framework of special relativity, allowing us to describe physical events in a unified way. A four-vector is an extension of the ordinary three-dimensional vectors to include time as a fourth dimension, formatted as $(ct,\mathbf{r})$, where $c$ is the speed of light, $t$ is the time component, and $\mathbf{r}$ represents the spatial components in three dimensions.

When considering physical phenomena at high velocities, close to the speed of light, the use of four-vectors becomes essential. They transform between different inertial frames through Lorentz transformations, ensuring that the physics remains consistent, regardless of the observer's state of motion. In the context of quantum mechanics, the concept of a four-vector is used to bridge the space-time description of particle events, such as the four-momentum vector, and wave-like properties, represented through the frequency vector in the de Broglie hypothesis.

Special Relativity

Special relativity is a theory postulated by Albert Einstein that revolutionized the understanding of space, time, and energy. It is built upon two postulates: the laws of physics are the same in all inertial frames, and the speed of light in a vacuum is constant for all observers, regardless of their relative motion.

In special relativity, time and space are not absolute; instead, they are intimately linked in a four-dimensional space-time continuum. Consequences of the theory include time dilation, length contraction, and the equivalence of mass and energy (as expressed in the famous equation $E=m{c}^2$). Special relativity plays a critical role in modern physics, underpinning both quantum mechanics and general relativity, and is key to understanding particle behavior at relativistic speeds.

Lorentz Transformation

The Lorentz transformation is a set of equations in special relativity that describes how, according to an observer, the position and time of an event change when moving from one inertial frame to another. The Lorentz transformation preserves the space-time interval between any two events, ensuring the invariance of physical laws across different reference frames.

This mathematical tool is necessary to transform the coordinates and times of events from one inertial frame to another. It assures that the speed of light remains constant in all inertial frames and accounts for the aforementioned phenomena such as time dilation and length contraction. In relation to the de Broglie wavelength and quantum mechanics, the Lorentz transformation ensures that certain quantities, like the phase velocity of a particle's de Broglie wave, remain unchanged under different frames of reference, reinforcing the concept of relativistic invariance in quantum systems.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides the description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces concepts that are quite different from classical physics, such as wave-particle duality, quantization, and uncertainty principles.

In quantum mechanics, particles such as electrons and photons have both wave-like and particle-like properties. The de Broglie hypothesis, central to quantum mechanics, proposes that any particle can be associated with a wave, and the de Broglie wavelength represents the wave aspect of the particle. Quantum mechanics and special relativity are interwoven into relativistic quantum mechanics, which describes quantum phenomena in a way that is consistent with the theory of special relativity. This field paves the way to an even more comprehensive theory of quantum field theory, which rigorously combines the principles of quantum mechanics with those of special relativity.