Question about the pedagogy of operators

[danieltanfh95]

how do you usually get introduced to position/momentum operators? like whats the physical intuition/motivation for it?

 

[martinbn]

How did you get introduced to them?

 

[PeterDonis]

how do you usually get introduced to position/momentum operators? like whats the physical intuition/motivation for it?

What textbooks have you looked at?

 

[QuarkyMeson]

You start with classical mechanics. In the Hamiltonian formulation, the state of a system is a point in phase space, given by position and momentum. Then you start seeing parallels between the classical and quantum descriptions.

The Poisson bracket $\{x, p\} = 1$ becomes the commutator $[\hat{x}, \hat{p}]= i\hbar$, and Hamilton’s equations become the Heisenberg equations of motion. In classical mechanics, momentum is the generator of spatial translations. In QM, the translation operator is $T(a) = exp(−ia\hat{p}/\hbar)$, which gives $\hat{p} = −i\hbar d/dx$ in the position representation. A de Broglie plane wave $exp(ipx/\hbar)$ is an eigenfunction of that operator with eigenvalue p.

Sakurai’s Modern Quantum Mechanics, Ch. 1, motivates p̂ from translations this way.

This is (mayhap unfortunately?) how it was introduced in my undergrad QM course.

 

[PeterDonis]

Sakurai’s Modern Quantum Mechanics, Ch. 1, motivates p̂ from translations this way.

Ballentine has a similar treatment, where he derives key operators from considerations of invariance under various symmetries.

 

[Matterwave]

Intuition was given in classical mechanics. In QM, we started with a wave function of a single particle in 1-D position representation: $\psi(x,t)$ and then was told $\hat{x}f=xf$ and $\hat{p}f=-i\hbar\partial_x f$. Finally we were given a commutator as $[\hat{A},\hat{B}]f=\hat{A}\hat{B}f-\hat{B}\hat{A}f$ and worked out the commutator for x and p.

Mostly we worked through examples. This is my (probably quite garbled) recollection of Griffiths QM text which is where these concepts were first introduced to me.

 

[Matterwave]

Ballentine has a similar treatment, where he derives key operators from considerations of invariance under various symmetries.

Did y'all start with Ballentine and Sakurai? My undergrad did Griffiths presumably because it's a (much) gentler introduction.

 

[QuarkyMeson]

Ballentine has a similar treatment, where he derives key operators from considerations of invariance under various symmetries.

Nice. We're using Ballentine for my grad QM course next semester. It will be interesting to take another go at QM when I'm also not trying to learn new notation and merely treading water for the exams.

 

[QuarkyMeson]

Did y'all start with Ballentine and Sakurai? My undergrad did Griffiths presumably because it's a (much) gentler introduction.

We started with Sakurai. As far as I've been told there were only two options the professor was considering, Shankar and Sakurai. He choose the latter.

I've heard others start with Griffiths, but his book seems universally loathed at my university.

 

[PeterDonis]

Did y'all start with Ballentine and Sakurai? My undergrad did Griffiths presumably because it's a (much) gentler introduction.

My introduction to QM in college was lecture notes by one of the professors who was teaching the course in nuclear physics I was taking (my major was Nuclear Engineering). I only encountered any of the standard textbooks years after leaving college, and only because of my personal interest in physics, not in connection with any kind of school. Of the three textbooks you mention, I came across Ballentine first, then Griffiths, then Sakurai (the last was as a result of a PF thread where it was referenced).

 

[danieltanfh95]

How did you get introduced to them?

I can't recall before, but recently I restarted from susskind's theoretical minimum, then read some of David Tong's notes. I'll see if I can find the textbooks mentioned somewhere.

I was trying to explain the physical intuition behind operators using only high school maths. This is the approach I'm going for

1. Let's assume that this system is a blackbox, and thus can be represented by some function psi(x). Treat it like a videogame boss that you don't know details off but can observe from the lootpool or the attack sequeunce.
2. SInce we can only interact with this blackbox through experiments, our results shows different probabilities for different results.
3. From historical experimentation, the connection between these probabilities and results is the born rule.
4. But now by treating the system as a probabilisitic one, we can use ordinary probabilities to find out stuff, as long as our detector provides ideal measurement (defined as so and so)
5. Operators are then defined as a specific operation we extract as a common structure from expectation values.

 

[Matterwave]

If you are trying to use the simplest possible math to motivate $\hat{x}$ and $\hat{p}$ perhaps just start from the 1-D position-representation wave function which gives a probability distribution: $\rho(x, t) = |\psi(x,t)|^2$.

We immediately get:
$$
\langle \hat{x}(t)\rangle = \int_{-\infty}^{\infty} \psi^* x \psi dx
$$
Which is just the definition of expectation value given a probability distribution. And we can use Ehrenfest's theorem (without naming it, and ignoring any circularity, since we are just motivating things lol) to get:

$$
\begin{aligned}
\langle \hat{p}(t)\rangle &= m\frac{d}{dt}\langle \hat{x}(t)\rangle \\
&= m\partial_t\int_{-\infty}^{\infty} x\psi^*\psi dx \\
&= m\int_{-\infty}^{\infty} x (\psi\partial_t \psi^* + \psi^*\partial_t \psi)
\end{aligned}
$$

Now we can apply the 1-D position-representation Schroedinger equation for a free particle (and its complex conjugate):
$$i\hbar\partial_t \psi(x,t) = -\frac{\hbar^2}{2m}\partial^2_x \psi(x,t)$$

To get (the negative comes from complex conjugation):
$$
\begin{aligned}
\langle \hat{p}(t)\rangle &= \frac{i\hbar}{2}\int_{-\infty}^{\infty} x (\psi^*\partial^2_x\psi - \psi\partial^2_x\psi^*) \\
&= \frac{i\hbar}{2}\int_{-\infty}^{\infty} x \partial_x(\psi^*\partial_x\psi - \psi\partial_x\psi^*)
\end{aligned}
$$

Integrate by parts to move the $\partial_x$ onto $x$ (to get 1) discard the boundary term. Then integrate by parts again to move the $\partial_x$ off the $\psi^*$ and again discard the boundary term.

We'll arrive at:

$$
\langle \hat{p}(t)\rangle = \int_{-\infty}^\infty \psi^* \left( \frac{\hbar}{i} \partial_x\right)\psi
$$

So now just extract the expression for $\hat{p}$.

It's just a motivation. Not a derivation with any rigor. But it's very simple, requiring only a bit of integration by parts (and of course, working with imaginary numbers which is kind of unavoidable in QM). I'm not sure it's quite High School level... maybe for an AP student it'll be ok.

I also neglected $V$ just so I could type less but it doesn't change this motivation. Terms involving it (assuming it's real) will cancel out.

A derivation starting from translation symmetry is of course much prettier.