# What's the equation $\hat{a}|0\rangle=0$ saying?

This is a well-known result of ladder operators, which obviously means that you can't remove energy from the vacuum. But what is $\hat{a}|0\rangle=0$ actually saying? How does it say the "you can't" part of the sentence? My best guess is "The probability of having such a state is $0^2=0$ ", but I'm not sure.

A satisfactory answer to this must contain the word "zero", since that's the only thing that is given to us by the formula. Sentences like "you can't do that" or "that doesn't exist" aren't good translations of what the formula is saying. The formula is saying that some quantity is zero, and my question is: What is that quantity?

• Isn't it more of you can't destroy particles that aren't there? Commented Jan 23, 2017 at 2:04
• But how does it say the "you can't" part? What does the 0 mean? Commented Jan 23, 2017 at 2:06
• See my answer to physics.stackexchange.com/q/112807 Commented Jan 23, 2017 at 2:09
• @JamalS I already saw your answer, but it seemed to me that you just demonstrated why it can't be |0>, not why it must be 0. Every mathematical expression has a physical translation, and "you can't" is too ambiguous for maths. Maths would say, for example, "the probability that such a state exists is 0", which is, as I said, my best (and only) guess. Commented Jan 23, 2017 at 2:15
• Doesn't is say "There is no such state arrived at by destroying the vacuum state"? Commented Jan 23, 2017 at 2:28

The formula says that the result of applying $a$ to the state $|0\rangle$ (which is a physical state) is $0$ (which is not a physical state). Since we know that $a$ is the destruction operator, what this equation implies is that there is no physical state which has less particles than $|0\rangle$; therefore, $|0\rangle$ is to be interpreted as a state with zero particles.
We say that "you can't take particles out of the vacuum" because if you attempt to calculate $a|0\rangle$, you get something which is not a physical state (for example, it's not normalizable) or, if you prefer, something that has zero overlap with every other vector.
• @PatoRaimundo: For any state $|\psi\rangle$ at all, the inner product of $0$ (not $|0\rangle$!) and $|\psi\rangle$ is zero. This means that no matter the initial state of your system, the probability to ever find it in the state $0$ is, well, $0$, so that state is inaccessible. Commented Jan 23, 2017 at 12:56
• Great! That's the answer I was looking for. The equation is saying: the state $\hat{a}|0\rangle=|?\rangle$ is a function or vector that is just zero everywhere. Therefore, for any state, the probability that they are the same is $\langle ?|\psi\rangle=\textbf{0}|\psi\rangle =0$, even with itself: $\langle?|?\rangle =0$. So it's saying "the probability of the system being is such a state is zero". If you could include something like that in your answer for more people to see, then I'll accept it. Commented Jan 24, 2017 at 8:30