The wave function and the Schrödinger equation

  • The Schrödinger equation plays the role of $F=ma$ to determine the dynamics (time evolution) of quantum entities.
  • Instead of the position, $x(t)$, of a particle, we will deal with a wave function, $\Psi(x,t)$.
  • The measurement problem: Wavefunction collapse?!

The wave function

In classical physics (for example, in Analytical Mechanics) we attempto find the position of a particle $x(t)$. But in quantum mechanics we attempt to find the wave function $\Psi(x,t)$ of a particle.

$\Psi$ is the amplitude of a field--just as we talk about the not-directly-observed electric field amplitude. Ripples (waves) in such a field are what interact creating all manner of interference phenomena.

$\Psi(x,t)$ is a solution of....

The (time-dependent) Schrödinger equation: $$i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\del^2\Psi}{\del x^2}+V(x,t)\Psi.$$

where $$\hbar=\frac{h}{2\pi}=1.054572\times10^{-34}{\text J s}.$$ The QM recipe is

  • Put in the $V$ (usually $V(x)$),
  • Solve the Schrödinger equation,
  • Apply boundary conditions.

Problem:   Assume that $\Psi$ is a solution of the Schrödinger equation. Let $A$ be a constant. Is $A\Psi$ also a solution?

The Schrödinger equation is linear if it fulfills these conditions:

  • If $\Psi_1$ and $\Psi_2$ are two linearly independent solutions,
  • then $A\Psi_1+B\Psi_2$ is also a solution if the equation is linear.

The consequence is that any linear combination (or "superposition") of solutions is also a solution.

Statistical interpretation of $\Psi(x,t)$

What *is* this wave function? The statistical interpretation (due to Max Born):

The probability of finding the particle between $x=a$ and $b$, at time $t$ is: $$\int_a^b |\Psi(x,t)|^2\,dx.$$

So, $|\Psi(x,t)|^2$ is a probability density. It has SI units of [probability]/meter.

Complex numbers

The wave function, $\Psi(x,t)$ is a complex number.

Complex numbers have the properties of 2-D vectors in a space in which:

  • The $x$ component of the vector is the real part of the number, and
  • the $y$ component of the vector is the imaginary part.

Show that for a complex number $z=a+bi$ ($a$ and $b$ are real numbers), the "magnitude", $|z|$, of a complex number--which is the length of it's 2-D vector representation--has the property that: $$|z|^2 = z\cdot z^*$$ where $|z|=\sqrt{a^2+b^2}$ is also called the measure or norm of the complex number, and $z^*=a-bi$ is the complex conjugate of $z$.

Watch out for ${}^*$: You're used to seeing it used as a multiplication sign. But complex conjugates are written with a superscript asterisk $()^*$. I will frequently write:

$$|\Psi|^2 \equiv \Psi^* \Psi.$$ This means "(the complex conjugate of $\Psi$) times $\Psi$"
or
"(the norm of the complex number $\Psi$) squared".

It never means "$\Psi$ times $\Psi$".

What does it mean that QM only tells us something statistical about the particle's position?

A detector shows us the position of a particle at a particular time $t$.

Let's call that position $x=C$, Philosophically there are (at least) 3 options:

  1. Realist: The particle existed at (was localized at) $C$ before the measurement took place. QM is good, but incomplete. A better theory will be able to calculate exactly where the particle is at any time. (This position is along the lines of Laplace's view of scientific determinism.)
  2. Orthodox (Copenhagen): the process of measurement summons a particle into existence, which had no definite location beforehand.

    In the double-slit experiment with the dim light: It's not that we're missing the flashes/reflections from the electrons that formed the interference pattern): There was no electron that could have been seen.

  3. Agnostic: Either one of the two options above might be true. Doesn't seem like there's any way to actually tell the difference. Maybe a better theory will come along. Meanwhile we should suspend judgement.

In 1964, John Bell ("Bell's inequality") showed that 1 and 2 have different, observable consequences. And experiments since then (Alain Aspect was one of the first to carry them out) have been in agreement with option 2.

Repeated measurements

The wave function representing the probability density, $\Psi(x,t)$ may be spread out in space.

If you measure a particle once at $C$, and then make another measure very quickly, where do you find the particle on the second measurement?

The particle is again found at $C$! (Sort of like the bowling ball you put down, look away from it, and then look back at it...)

But the wave function was spread out in space. Why doesn't a subsequent measurement find the particle at one of the other locations where it also had a finite probability?

The Copenhagen interpretation deals with this by claiming that, upon making a measurement of the position of a particle, the wave function must collapse [to a delta function] in order that an immediately following measurement will find the particle at the same place again.



As time passes, $\Psi(x,t)$ might again spread out...

Wave collapse is the nub of the measurement problem

There are two problems. One is that quantum mechanics, as it is enshrined in textbooks, seems to require separate rules for how quantum objects behave when we’re not looking at them, and how they behave when they are being observed. When we’re not looking, they exist in “superpositions” of different possibilities, such as being at any one of various locations in space. But when we look, they suddenly snap into just a single location, and that’s where we see them. We can’t predict exactly what that location will be; the best we can do is calculate the probability of different outcomes.

The whole thing is preposterous. Why are observations special? What counts as an “observation,” anyway? When exactly does it happen? Does it need to be performed by a person? Is consciousness somehow involved in the basic rules of reality? Together these questions are known as the “measurement problem” of quantum theory.

Sean Carroll, Even physicists don't understand quantum mechanics

[The second problem: We don't really know what the wave function is...]

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Image credits

D. Jackmanson, Samad Jee