Lecture 3 - Quantum Mechanical Wave Function

As some announcements, some of the stuff in the reading is very much unknown. That's okay! For the book, it's more linear, but if you're following the textbook reading primarily, that's all good.

You'll need to do:

Read 2.2, 2.5, 2.6 (not through 2.6.1)
Read the lecture slides on Quantum Superposition
HW 1 (due next week) (the first WA is next friday)

2.1: Wave-Particle Duality (1.5.3)

In physics, we look at things in experiments, then model it via mathematics.

Evidence for the wave model and against the photon model of light occurred with Young's Double slit experiments:

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If light behaves like a wave, then some of the light constructively interferes, while some of it constructively interferes. As such, then this gave evidence that light acted like a wave. Either the light goes through the slit, or it doesn't. The single slit shows the standard wavelike properties of light as a wave, but with the double slit we didn't see two single slit things. As a result, light acted more like a wave.

But there was also evidence for the photon model instead. The Photoelectric effect was shown by kicking out electrons shot from metal by photons of some color/energy. It seemed that there was some relationship between photon color and energy.

So we have two experiments that seem to contradict each other. So our model (either one) is incomplete, so then we have to modify them to work together. It wasn't that the model wasn't useful, it's just not complete (not quite right).

The Quantum Mechanical Wave Function

This was the new model. this works for all QM (very small particles, less than 1 billion particles) systems. Not just light!

The wave function model of light is where light consists of particles (photons). However, the photons only exist when they are detected. Their detection probability is calculated from their probability amplitude (wave function), which propagates (and interferes) like a wave.

If you integrate the square of it over the range of your measurement (the volume of the measurement), then you get a probability. In this model, it is the detector that brings the particle into existence.

Note

Here, a detector is a very loose term. Is a sensor a detector? Are our eyes? Is it the random air particle in the room? We'll later differentiate when and where this distinction comes up.

It doesn't even have to be light though! We can do it with 40 electrons sent, one by one, and get the same result:

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Notice the wave patterns as we add more and more electrons.

The Math

We define $Ψ (\vec{r}, t)$ as the wave function. $Ψ$ has a codomain of $C$ . We note that it is not the probability itself. Rather:

| Ψ (\vec{r}, t) |^{2}

is the probability density of finding the particle at $\vec{r}$ . As an aside:

| Ψ (\vec{r}, t) |^{2} d V

is the probability of finding a particle in volume $d V$ .

Review of $C$

Some theorems:

z = a + b i, i = \sqrt{- 1}, a = R (z), b = I (z)

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Here we have:

z = r \cos (θ) + i r \sin (θ) = r e^{- i θ}

Euler's formula is when $r = 1$ . Here:

r = \sqrt{a^{2} + b^{2}}, θ = \arctan (b / a)

Note that:

e^{α} e^{β} = e^{α + β}

\cos (ϕ) = \frac{e^{i ϕ} + e^{- i ϕ}}{2}, \sin (ϕ) = \frac{e^{i ϕ} - e^{- i ϕ}}{2}

More theorems:

z^{*} = x - i y, z^{*} = r e^{- i θ}

| z |^{2} = z^{*} z \neq z^{2}

|z|^2 = z^* z = a^2 + {#2} = r^2

Ψ = Ψ_{r e a l} + i Ψ_{i m a g}, | Ψ |^{2} = Ψ^{*} Ψ = Ψ_{r e a l}^{2} + Ψ_{i m a g}^{2}

2.1: Wave-Particle Duality (1.5.3)

The Quantum Mechanical Wave Function

The Math

Review of C

Review of $C$