De Broglie Slide

PHY-1109 Physics This response provides the complete content regarding De Broglie Waves, Wave Function, and Wave Velocity, copied verbatim from the source materials, with figure/image placement noted.

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De Broglie Waves

A moving body behaves in certain ways as though it has a wave nature. A photon of light of frequency $v$ has the momentum $p=\frac{hv}{c}=\frac{h}{\lambda }$ Since, $c=v\lambda$. The wavelength of a photon is therefore specified by its momentum according to the relation. $\lambda =\frac{h}{p}\text{ (1)}$ De Broglie suggested that Eqn. (1) is a completely general one that applies to material particles as well as to photons. The momentum of a particle of mass $m$ and velocity $v$ is $p=mv$ and its de Broglie wavelength is accordingly $\lambda =\frac{h}{mv}\text{ (2)}$ The greater the particle’s momentum, the shorter its wavelength. In Eqn. (2) $m$ is the relativistic mass $m=\frac{m_0}{\sqrt{1-v^2/c^2}}$ As in the case of EM waves, the wave and particle aspects of moving bodies can never be observed at the same time. We therefore cannot ask which is the “correct” description. All that can be said is that in certain situations a moving body resembles a wave and in others it resembles a particle.

Which set of properties is most conspicuous depends on how its de Broglie wavelength compares with its dimensions and the dimensions of whatever it interacts with. De Broglie had no direct experimental evidence to support his conjecture. However, he was able to show that it accounted in a natural way for the energy quantization—the restriction to certain specific energy values—that Bohr had to postulate in his 1913 model of the hydrogen atom. Within a few years Eqn. (2) was verified by experiments involving the diffraction of electrons by crystals.

Let us look into the question of what kind of wave phenomenon is involved in the matter waves of de Broglie.

The Wave Function $\Psi$

In water waves, the quantity that varies periodically is the height of the water surface. In sound waves, it is pressure. In light waves, electric and magnetic fields vary. What is it that varies in the case of matter waves?.

The quantity whose variations make up matter waves is called the wave function, symbol $\Psi$ (the Greek letter psi). The value of the wave function associated with a moving body at the particular point $x,y,z$ in space at the time $t$ is related to the likelihood of finding the body there at the time.

The wave function $\Psi$ itself, however, has no direct physical significance. There is a simple reason why $\Psi$ cannot by interpreted in terms of an experiment. The probability that something be in a certain place at a given time must lie between 0 (the object is definitely not there) and 1 (the object is definitely there). An intermediate probability, say 0.2, means that there is a 20% chance of finding the object. But the amplitude of a wave can be negative as well as positive, and a negative probability, say 0.2, is meaningless. Hence $\Psi$ by itself cannot be an observable quantity.

This objection does not apply to $|\Psi {|}^2$, the square of the absolute value of the wave function, which is known as probability density. The probability of experimentally finding the body described by the wave function $\Psi$ at the point $x,y,z$, at the time $t$ is proportional to the value of $|\Psi {|}^2$ there at $t$. A large value of $|\Psi {|}^2$ means the strong possibility of the body’s presence, while a small value of $|\Psi {|}^2$ means the slight possibility of its presence. As long as $|\Psi {|}^2$ is not actually 0 somewhere, however, there is a definite chance, however small, of detecting it there. This interpretation was first made by Max Born in 1926.

There is a big difference between the probability of an event and the event itself. Although we can speak of the wave function $\Psi$ that describes a particle as being spread out in space, this does not mean that the particle itself is thus spread out. When an experiment is performed to detect electrons, for instance, a whole electron is either found at a certain time and place or it is not; there is no such thing as a 20 percent of an electron. However, it is entirely possible for there to be a 20 percent chance that the electron be found at that time and place, and it is this likelihood that is specified by $|\Psi {|}^2$.

De Broglie Wave Velocity

How fast do de Broglie waves travel?. Since we associate a de Broglie wave with a moving body, we expect that this wave has the same velocity as that of the body. Let us see if this is true.

If we call the de Broglie wave velocity $w$, we can apply the usual formula $w=v\lambda$ to find $w$. The wavelength is simply the de Broglie wavelength $\lambda =\frac{h}{mv}$ We shall take the frequency $v$, to be that specified by the quantum equation $E=hv$ Hence $v=\frac{E}{h}$ Or since $E=m{c}^2$ We have $v=\frac{m{c}^2}{h}$ The de Broglie wave velocity is therefore $w=v\lambda$ $w=\frac{m{c}^2}{h}\frac{h}{mv}=\frac{c^2}{v}\text{ (3)}$ Because the particle velocity $v$ must be less than the velocity of light $c$, the de Broglie waves always travel faster than light!. In order to understand this unexpected result, we must look into the distinction between phase velocity and group velocity. Phase velocity is what we have been calling wave velocity.

Mathematical Description of Waves

Let us begin by reviewing how waves are described mathematically. For simplicity we consider a string stretched along the $x$ axis whose vibrations are in the $y$ direction, as in Figure, and are simple harmonic in character.

Image/Diagram Note: Figure illustrating a Vibrating string (showing $y$ vs. $x$ at $t=0$) and how the displacement varies with time ($y=A\cos 2\pi vt$ at $x=0$).

If we choose $t=0$ when the displacement $y$ of the string at $x=0$ is a maximum, its displacement at any future time $t$ at the same place is given by the formula $y=A\cos 2\pi vt\text{ (4)}$ where $A$ is the amplitude of the vibrations (that is, their maximum displacement on either side of the $x$ axis) and $v$ their frequency. Eqn. (4) tells us what the displacement of a single point on the string is as a function of time $t$.

Hence the displacement $y$ of the string at $x$ at any time $t$ is exactly the same as the value of $y$ at $x=0$ at the earlier time $t=x/w$. By simply replacing $t$ in Eqn. (4) with $t=x/w$, then, we have the desired formula giving $y$ in terms of both $x$ and $t$: $y=A\cos 2\pi v(t-\frac{x}{w})\text{ (5)}$ As a check, we note that Eqn. (5) reduces to Eq. (4) at $x=0$. Eqn. (5) may be rewritten $y=A\cos 2\pi (vt-\frac{vx}{w})$ Since $w=v\lambda$ We have $y=A\cos 2\pi (vt-\frac{x}{\lambda })\text{ (6)}$ Eqn. (6) is often more convenient to use than Eqn. (5).

Perhaps the most widely used description of a wave, however, is still another form of Eqn. (5). The quantities angular frequency $\omega$ and wave number $k$ are defined by the formulas $\omega =2\pi v\text{ (7)}$ $k=\frac{2\pi }{\lambda }\text{ (8)}$ or $k=\frac{\omega }{w}\text{ (9)}$ The unit of $\omega$ is the radian per second and that of $k$ is the radian per meter.

Angular frequency gets its name from uniform circular motion, where a particle that moves around a circle $v$ times per second sweeps out $2\pi v\text{ rad/s}$. The wave number $k$ is equal to the number of radians corresponding to a wave train 1 m long, since there are $2\pi \text{ rad}$ in one complete wave. In terms of $\omega$ and $k$, Eqn. (5) becomes $y=A\cos (\omega t-kx)\text{ (10)}$ In three dimensions $k$ becomes a vector $\mathbf{k}$ normal to the wave fronts and $x$ is replaced by the radius vector $\mathbf{r}$. The scalar product $\mathbf{k}\cdot \mathbf{r}$ is then used instead of $kx$ in Eq. (10).

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Now that you have all the raw content, we can move on to the next phase of learning. What aspect of De Broglie’s hypothesis—perhaps the unexpected result of $w>c$, or the concept of the wave function—would you like to discuss first?