How do we get the expectation value formula?

1. Oct 11, 2013

Outrageous

<x>= ∫ complex ψ x ψ dx
How do we get this formula? And why must the complex ψ must be placed in front?
Please guide or any link to help,not really understand this makes me difficult to start in quantum mechanics.
Your help is really appreciated. Pls

2. Oct 11, 2013

Staff: Mentor

Observation, it fits to the measurement results (there are also some theoretical arguments that it cannot be everything, but those are details. The main point is: the theory was constructed to match observations).

Do you mean complex conjugation? It does not have to, you multiply complex numbers here.

3. Oct 11, 2013

Outrageous

Thanks. One more to ask, even schrodinger equation is also constructed to match observation?

4. Oct 11, 2013

kaplan

All of physics is - observation, plus mathematical consistency (and a certain degree of "taste").

You can derive the Schrodinger equation from some more basic assumptions: that physical states are elements in a vector space, and that time evolution acts on that vector space as a unitary transformation. Then given the relation between energy and time in classical physics, the Schrodinger equation follows.

5. Oct 12, 2013

Outrageous

Thanks~ Do you have any link or book recommended for the derivation mentioned?

6. Oct 12, 2013

kaplan

Chapter 3 of Ballentine is OK. There are probably better references (overall that book isn't very good), but that's the only one I can think of off the top of my head.

7. Oct 12, 2013

Outrageous

Thanks, I try to check it out^^

8. Oct 12, 2013

Staff: Mentor

There is also an analogy to probability theory, based on the Born rule that $\psi^* \psi$ is a probability distribution. The expression of the quantum-mechanical expectation value
$$\langle \hat{A} \rangle = \int \psi^* \hat{A} \psi d\tau$$
is analogous to the expectation value for a continuous random variable
$$E[X] = \int_{-\infty}^{\infty} x f(x) dx$$
especially in the case you mention where you want $\langle x \rangle$ for $\psi(x)$.

9. Oct 13, 2013

Staff: Mentor

You might be interested in checking out Gleason's Theorem:
http://kof.physto.se/theses/helena-master.pdf [Broken]

For the basis of Schrodinger's Equation see (as has already been mentioned) Chapter 3 - Ballentine - Quantum Mechanics - A Modern Development:
http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf

Believe it or not it actually follows from symmetry - strange but true.

Actually so does Classical Mechanics for that matter - and how it is connected to QM is very interesting - see Landau - Mechanics for example.

In fact this is one of the very deep revelations of physics and well worth becoming acquainted with:
http://www.pnas.org/content/93/25/14256.full.pdf

All books on QM suck in their own unique way and its a matter of taste what approach you take to and book you gravitate towards. I personally love Ballentine - to me it was a revelatory book on QM - but opinions vary. Check it out for yourself and make up your own mind.

Thanks
Bill

Last edited by a moderator: May 6, 2017