I know that the Bandwidth Theorem (BT) and the Heisenberg Uncertainty Principle (HUP) are basically the same thing, and stem from the fact that for operators $A,B$, we have:

$$\Delta A \Delta B \geq \frac{1}{2} | \langle \psi |[A,B]|\psi\rangle.$$

The HUP is obtained using the fact that $[x,p] = i\hbar$. In my class, we were asked to prove this inequality, but were given without proof the value of $[x,p]$.

I've now been asked to prove the bandwidth theorem, which states that

$$\Delta f \Delta t \geq \text{any positive constant}$$

I assume that $f$ is frequency and $t$ is time. It seems that this theorem is folklore, because I haven't been able to find any rigorous proofs. It seems intuitive, especially after watching this 3B1B video, however I have some questions about how to prove this:

1. What even are $f$ and $t$? Functions? I know that the formula for $\Delta A^2$ is given by $\langle \psi | A^2 |\psi\rangle -\langle \psi | A |\psi\rangle^2 $. It's unclear how to compute this for $f$ or $t$, since I don't see how they can "operate" on $\psi$. I'm aware that we can use $$\langle f_1(x)|f_2(x)\rangle := \int_\mathbb{R} \bar{f_1}(x)f_2(x) dx$$ as an inner product.
2. Similarly, how was $[x,p] = i\hbar$ obtained?

I'm sure that the answer has something to do with a Fourier series, but my knowledge of them is shaky (basically just know the definitions), and most discussions of this theorem that I've found have been handwavy to say the least.

For what it's worth, I would still like to figure this problem out for myself (I'm not asking for a full solution), but I need to understand more about what the BT is actually stating before I can try to prove it.

I know that the Bandwidth Theorem (BT) and the Heisenberg Uncertainty Principle (HUP) are basically the same thing, and stem from the fact that for operators $A,B$, we have:

$$\Delta A \Delta B \geq \frac{1}{2} | \langle \psi |[A,B]|\psi\rangle.$$

The HUP is obtained using the fact that $[x,p] = i\hbar$. In my class, we were asked to prove this inequality, but were given without proof the value of $[x,p]$.

I've now been asked to prove the bandwidth theorem, which states that

$$\Delta f \Delta t \geq \text{some positive constant}$$

I assume that $f$ is frequency and $t$ is time. It seems that this theorem is folklore, because I haven't been able to find any rigorous proofs. It seems intuitive, especially after watching this 3B1B video, however I have some questions about how to prove this:

1. What even are $f$ and $t$? Functions? I know that the formula for $\Delta A^2$ is given by $\langle \psi | A^2 |\psi\rangle -\langle \psi | A |\psi\rangle^2 $. It's unclear how to compute this for $f$ or $t$, since I don't see how they can "operate" on $\psi$. I'm aware that we can use $$\langle f_1(x)|f_2(x)\rangle := \int_\mathbb{R} \bar{f_1}(x)f_2(x) dx$$ as an inner product.
2. Similarly, how was $[x,p] = i\hbar$ obtained?

I'm sure that the answer has something to do with a Fourier series, but my knowledge of them is shaky (basically just know the definitions), and most discussions of this theorem that I've found have been handwavy to say the least.

For what it's worth, I would still like to figure this problem out for myself (I'm not asking for a full solution), but I need to understand more about what the BT is actually stating before I can try to prove it.