Phase of an Amplitude Function

[chessforce]

In Feynman's Lectures on Physics, he constantly refers to the phase when discussing quantum mechanical amplitudes, but does not elaborate on what the phase means physically (e.g. the way the square of the amplitude gives the probability of finding a particle somewhere). So, if someone can explain the significance of the phase, that would be great. Thanks!

 

[Mentz114]

The phase actually has no accepted physical meaning. It comes into play when we add amplitudes, so interference takes place. When we square ( find the modulus of) a complex number the phase is irrelevant, so it never affects the last squaring process.

The amplitude is a complex number of the form (A is real)

$$A(x,t)e^{\frac{iE}{h}(x.k - \omega t)}$$

the square its modulus is $$A(x,t)^2$$, regardless of the term in the exponential which is dimensionless and is taken as an angle - hence 'phase'.

You can also multiply any amplitude by $$e^{i\theta}$$ without affecting it's modulus. So if every wave function in the universe is multipied by that factor - nothing changes. This is called global phase invariance.

 

[CarlB]

In the theory of small amplitude waves, one approximates a wave by a sine wave. To describe a sine (or cosine) wave, you need an amplitude A and a phase $$\delta$$:

$$\psi(x) = A \cos(x + \delta) = R( A \exp(i(x+\delta)))$$

where "R" means take the real part.

When two such waves travel through the same media, they add together the obvious way $$\psi = \psi_1 + \psi_2$$. Using trigonometry, this becomes

$$\psi_1(x) + \psi_2(x) = A_1 \cos(x + \delta_1) + A_2 \cos(x + \delta_2)$$
$$= R(A_1 \exp(i(x+\delta_1)) + A_2\exp(i(x+\delta_2)))$$

Another way of saying the same thing is that (small) real waves and complex waves add together the same way. It turns out to be easier to deal with the complex waves.

When we are discussing real things, like earthquake waves, it makes sense to use the real form and describe everything with sine waves. However, at this time, no one has determined a "real" meaning for the waves of quantum mechanics. So we might as well use the complex form. And since we can take the real part at any point in the computations, we might as well ignore it completely and rewrite the formulas to act as if the quantum waves are fundamentally complex.

Let me ignore the delta phase from here on. I've written the above using cos(x). To convert it to something more realisitic, you need to add a dependency on time t. For a wave moving steadily in the +x direction at speed c, one would want cos(x-ct).

When you write a wave as a function of space and time, it becomes clear that the real description of a wave must be missing something. That is, the real wave cos(x-ct) has information about the offset at the position x at time t, but it does not have information about the momentum at that spot.

With any sort of harmonic motion, the momentum is zero when the position is maximum. That is because when the position is maximum, it is getting ready to turn around and go the other way. This means that the momentum wave has to look like sin(x-ct). And when we talk about a real wave cos(x-ct), we also need to remember that there is also a momentum wave sin(x-ct). When we deal with a complex wave
$$\exp(x-ct) = \cos(x-ct) + i\sin(x-ct)$$
we are bundling the position and momentum information into the same object.

What quantum mechanics does is similar to this but everywhere I've written "position" and "momentum" in the above are not the position and momentum of QM. Instead, they are just the mathematics needed to make the wave oscillate correctly. In QM, position and momentum are operators, the wave function is just something that encodes the distribution of possible positions and momenta as a probability distribution, sort of. But the above analogies are why it's called "phase" in QM, I think.

 

[Anonym]

if someone can explain the significance of the phase, that would be great.

When you try to describe the results of your observations in the laboratory mathematically, you will find that they require use of two independent functions of space-time coordinates even for the simplest QM system – single electron/photon. Therefore, you should use 2-dim algebra, namely,
F(x,t)=a(x,t)+i*b(x,t). It may be presented alternatively in the following form:

F(x,t)=A(x,t)*exp(i*phi(x,t)).

A(x,t) is called amplitude and phi(x,t) is called phase (during last 200 years).In addition, if you want not to fool yourself, remember that English, Chinese or Hebrew are not tools suitable to describe physics.

Regards,Dany.

 

[chessforce]

Thank you very much for all the responses! They were very enlightening.