Quantum mechanics, uncertainty and measurement

Question

I'm trying to learn the very basics of quantum mechanics and trying to grasp the concepts of uncertainty.

My claim, which may be false is; the energy of a particle, lets say in a box, may be any value between the ground state and some energy we define as maximum in this case, lets say $E_2$. But, when we choose to measure it, the energy of the particle is exactly one of the quantum states.

I don't think this sounds correct, but if it's not I get the following problem: If the energy of the particle is discrete even when we don't measure it, the uncertainty, may not make sense. Let's say the uncertainty is $\Delta E<E_2-E_1$. Then, if the energy is $E_1 + \Delta E$, we would know that the energy is exactly $E_1$. Hence the uncertainty would be zero.

There is no doubt that I got something wrong here, and I would appreciate any help to make it clear what is wrong and what is correct, if any of it.

Answer 1

What do you mean by

"Then, if the energy is $E_1+\Delta E$"

? The energy of what? If you mean the measured energy, then it is impossible, since the measured energy can be just one of the allowed energies, in this case $E_1$ or $E_2$. If you mean the expected value of the energy before the measurement, then your conclusion is wrong, and I'll try to explain why.

First of all, you didn't define what you mean by uncertainty, so I'll assume that you are talking about the most used form for uncertainty in quantum mechanics - the standard deviation, which is what appears in the uncertainty principle.

There is no problem with the uncertainty being smaller than the energy difference between the discrete energy state.

First, to make sure you understand, the uncertainty is a result of the principle of superposition (whereas the uncertainty principle is a result of the non-orthogonality of eigenstates of different observables). If we speak about energy, then the uncertainty is zero if the particle (or whatever) is in an energy eigenstate of the system. But if the particle's state is a superposition of energy eigenstates, then the outcome of an energy measurement can be any of these energies, and therefore there is uncertainty in energy.

How much uncertainty? this depends on the coefficients of the superposition. The (squared modulus) of these coefficients are the probabilities to measure the corresponding energy (assuming for now that the energies are non-degenerate). Let's take an example with two energies, $E_1=0$ and $E_2=1$ (in whatever units you choose), and probabilities of $p$ for the first outcome and $1-p$ for the second. Then the expected value of the energy measurement is $pE_1+(1-p)E_2=1-p$. And the standard deviation is $\sqrt{p(1-p)}$. This standard deviation is the uncertainty we are talking about. And notice that it can take any value between $0$ (when $p=0$ or $p=1$) and $0.5$ (when $p=0.5$). Specifically, it can be very small, much smaller then the energy difference between $E_1$ and $E_2$. But for the difference to be very small, $p$ needs to be very close to $1$ or $0$. This means that with high certainty, the energy will be $E_1$ (in the former case), but there is still a small chance to measure $E_2$. The small uncertainty indicates exactly this.

Bottom line: the correct conclusion to draw if the uncertainty $\Delta E$ is small compared to differences in the discrete energy spectrum, is that it is more likely for the measured energy to be one of the energies in the spectrum than others.

Answer 2

The uncertainty principle (UP) has nothing to do with uncertainty.

In classical physics, a system is represented by some set of numbers like (x,y,z) or whatever. Each of these numbers have some particular value at a given time. And you can measure each of these numbers to any accuracy you like and put the classical system back into the state it was in before.

Quantum mechanics isn't like that. A quantum system is described by a set of operators A,B,C etc that are called observables. These operators are not a single number that you can measure and reset. Rather, the operators represent all of the possible outcomes of a measurement and together with the state give the probability that you will get each of the possible values if you do a measurement.

People often talk about probability as having something to do with uncertainty. This is nonsense. Probabilities have precise numerical values that we can calculate and you can't get such values from lack of knowledge. Rather, probabilities have to be numbers that are explained in some way by the underlying physics as being relevant for real experiments, see:

https://arxiv.org/abs/1508.02048.

The UP relates the probability distribution for one observable A to the probability distribution of another observable B. In particular, the UP says that if the probability distribution of A has a very high probability of being at a particular value, the probability distribution for B must be spread out according to a particular formula, and the same is true with A swapped for B. This doesn't mean that the probability distribution for A must be spread out over many values, but if it is not spread out then the probability distribution for B must be spread out over many values.