# I need the dirac notation expectation value explaining to me please?

• jeebs
In summary, the conversation discusses the expectation value in quantum mechanics and how it relates to the eigenvalues of an operator expressed in its orthonormal eigenbasis. The speaker also mentions a potential mistake in equating the coefficients of a wavefunction in this basis to the eigenvalues of the operator. They seek clarification on this topic to avoid similar mistakes in the future.
jeebs
Hi,

I find a lot of the time in QM i have been calculating things blindly. Take the expectation value for instance. I have worked this out in integral form plenty of times, but haven't really understood why I'm doing what I'm doing. I looked up wikipedia and apparently, for a measurable quantity/Hermitian operator A & a system in the state $$\psi$$, the expectation value is $$<A> = <\psi|A|\psi>$$. As I understand it, the expectation value of a measurement is the mean value if we repeated the measurement loads of times. Given this definition, it is not obvious to me why the equation above works. I don't see that just from looking at that equation, can anyone explain?

Anyway, I will take it for granted that $$<A> = <\psi|A|\psi>$$. I know that if the operator A is expressed matrix-style using its orthonormal eigenbasis $$|\phi_j>$$, then the eigenvalues aj found from the eigenvalue equation $$A|\phi_j> = a_j|\phi_j>$$ will appear on the leading diagonal. My notes also state that for a complete orthonormal basis, any vector $$|\psi>$$ can be written as $$|\psi>=\sum_j c_j|\phi_j>$$. I think it's okay to say that in this particular basis, cj = aj, so that we have $$A|\psi> = A\sum_j a_j|\phi_j> = \sum_j a_jA|\phi_j> = \sum_j a_j^2|\phi_j>$$. I then stick the bra on the other side, to make $$<A> = <\psi|A|\psi> = <\psi|\sum_j a_j^2|\phi_j> = \sum_j a_j^2<\psi|\phi_j>$$.
Apparently this is wrong though, wikipedia states that $$<A> = <\psi|A|\psi> =\sum_j a_j|<\psi|\phi_j>|^2$$.

I cannot see why that is or where I have went wrong, but I get the feeling I must be confused over something really fundamental and simple. If anyone could clear this up I'd really be grateful, it would probably stop me making mistakes in other areas as well.
Thanks.

Last edited:
jeebs said:
Hi,

I know that if the operator A is expressed matrix-style using its orthonormal eigenbasis $$|\phi_j>$$, then the eigenvalues aj found from the eigenvalue equation $$A|\phi_j> = a_j|\phi_j>$$ will appear on the leading diagonal.

That should immediately tell you that you can write

$$A=\sum_i a_i| \phi_i\rangle\langle \phi_i |$$

My notes also state that for a complete orthonormal basis, any vector $$|\psi>$$ can be written as $$|\psi>=\sum_j c_j|\phi_j>$$. I think it's okay to say that in this particular basis, cj = aj

Why would you think that the coefficients of your wavefunction in this eigenbasis are the same as the eigenvalues of your operator? Certainly that won't be true for every wavefunction, will it?

Apparently this is wrong though, wikipedia states that $$<A> = <\psi|A|\psi> =\sum_j a_j|<\psi|\phi_j>|^2$$.

I cannot see why that is

Simple,

\begin{aligned}\langle \psi|A|\psi\rangle &= \langle \psi|\left(\sum_i a_i| \phi_i\rangle\langle \phi_i |\right)|\psi\rangle \\ &= \sum_i a_i\langle \psi| \phi_i\rangle\langle \phi_i |\psi\rangle \\ &= \sum_i a_i\left|\langle \psi| \phi_i\rangle\right|^2\end{aligned}

thanks man

gabbagabbahey said:
Why would you think that the coefficients of your wavefunction in this eigenbasis are the same as the eigenvalues of your operator? Certainly that won't be true for every wavefunction, will it?

this is exactly the sort of daft mistake I need to iron out.

## 1. What is the Dirac notation?

The Dirac notation, also known as bra-ket notation, is a mathematical notation used to represent vectors and operators in quantum mechanics. It was developed by physicist Paul Dirac and is written in the form of <A|B>, where A and B are mathematical objects such as vectors or operators.

## 2. What is an expectation value in quantum mechanics?

In quantum mechanics, an expectation value is the average value of a physical quantity that is predicted by a quantum system. It is calculated by taking the inner product of the state vector with the corresponding operator and then normalizing it.

## 3. How is the expectation value calculated using Dirac notation?

The expectation value using Dirac notation is calculated by taking the inner product of the state vector with the corresponding operator. This is written as <A|B|A>, where A is the state vector and B is the operator. The result of this calculation is the expectation value of the operator B in the state A.

## 4. Can you give an example of calculating the expectation value using Dirac notation?

For example, let's say we have a quantum system described by the state vector |A> and we want to calculate the expectation value of the operator B in this state. The calculation would be written as <A|B|A>. If the result of this calculation is 3, then the expectation value of B in the state A is 3.

## 5. Why is Dirac notation useful in quantum mechanics?

Dirac notation is useful in quantum mechanics because it simplifies and streamlines calculations involving vectors and operators. It also allows for a more intuitive understanding of quantum systems and their properties. Additionally, it is a universal notation that is used by physicists and mathematicians around the world, making it easy to communicate and share ideas.

Replies
9
Views
1K
Replies
1
Views
765
Replies
5
Views
2K
Replies
4
Views
4K
• Quantum Physics
Replies
2
Views
1K
• Quantum Physics
Replies
5
Views
1K
Replies
5
Views
3K
Replies
4
Views
2K