Electron Waves, and the de Broglie Wavefunction

Prior to the developments of Born and Schrondinger, de Broglie posed a formulation for the wavefunction of a free-particle. In wave mechanics, it is most common to describes waves in terms of a frequency and a wavelength, and through various relationships, the wave-number and velocity, but for a particle, to what do these descriptions specifically relate? According to (my favourite bug-bear in quantum mechanics, but only in name) wave-particle duality, particles are in fact waves, as is evidenced by diffraction experiments with particles, and a feature which arose from such experiments, is a dependance upon certain features of diffraction and the velocity of the particles used. In fact, it turns out, that the diffraction phenomena does not depend, directly, upon velocity, but a particles' momentum and energy.

In 1924, de Broglie proposed that the wavelength of a free-particle (ie., one with no forces acting upon it) is inversely proportional to its momentum. The key expression, known as the de Broglie wavelength, which illustrates a decreasing wavelength with increasing speed, is,

`lambda_(dB)=h/p`

where h is Plancks constant and p is momentum

It is through this relationship that describing a 'particle' by a `wavefunction` makes sense. Using Schrodingers formulation, we define a wavefunction with the notation `Psi(barr,t)`

In quantum mechanics, the wave-function is taken to be the most complete description of the system. As such, the simple wavefunction below (and described here) cannot completely describe the system, for example, if the wave were to be measured at a particular place and time such that the amplitude at that precise instant were y=0, then we would know nothing about past or future evolution of the wave- or infact, if the point represents a wave at all!

`Psi(x,t)=Acos(kx-omegat+phi)`

Equation 1

Clearly, this is not a complete description of the system. The problem can be overcome by also describing the wave in terms of its derivative, which provides an indication as to the direction the amplitude will take, for example, the gradient at which the wave is changing.

The wave function expressed by Equation 1 is completely real valued. That is, there is no Imaginary component. In actual fact, waves are complex valued, meaning they have an Imaginary component in addition to the real component. This Imaginary component completes the description of the system, by accounting for the rate of change of the wavefunction.

A complex number is one of the form, `z=a+ib`, where `a` is the real component, and `ib` is the Imaginary component. Thus it goes to figure, that a wavefunction described by a complex function is also of this form. As alluded to, the Imaginary component describes the gradient and, as such, the Imaginary component is related to the real component through differentiation of the trigonometric term. Thus we can elaborate on Equation 1,

`Psi(x,t)=A[cos(kx-omegat+phi)+isin(kx-omegat+phi)]`

Equation 2

Now, exploiting the relationship between sinusoids and the exponential function, we can contract this definition, expressing the wavefunction as an exponential function,

`Psi(x,t)=Aexp(i(kx-omegat+phi)))`

At this point it is informative to express the descriptors, `k` and `omega`, the wave-number and angular frequency, respectively, in terms of particle-like properties. For example, using the definition of the de Broglie wavelength, we can express the wave-number (discussed here), `k=(2pi)/lambda_(dB)`, thus,

`k=(2pip)/h=p/hbar`

Where we have introduced the reduced Planck constant, `hbar`,

`hbar=h/(2pi)`

Similarly, using the Planck equation, `E=hf`, we can express the angular-frequency in terms of an energy,

`omega=2pif=(2piE)/h=E/hbar`

Hence, in terms of a particle energy and momentum, we can represent the de Broglie wave-function,

`Psi(x,t)=Aexp(i(p/hbarx-(E)/hbart+phi))`

Now considering we are dealing with a particle, we can describe the kinetic energy, E, in terms of its momentum, `p=mv`. Thus,

`E=1/2 mv^2=p^2/(2m)`

From the definition of the wave-number, we can express the wave's momentum, and from the angular frequency, the wave's energy,

`p=hbark` and `E=hf=hbaromega`

Hence, we can express the kinetic energy in terms of `k` and `omega`,

`hbaromega = (hbark)^2/(2m)`

We note that the left hand side involves a function of t, and the right hand side, a function of x. The significance of this becomes evident when considering the form of the Schrodinger wave-equation, and consequently, the energy expressed above is also an eigenvalue.

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