Expectation Values: Formula & Derivation

In summary, the formula for the expectation value in quantum mechanics is derived through a similar method as in statistics, but with the use of operators and kets in a Hilbert space. This leads to a "sandwich" method where the observable is expressed in the position basis and then integrated with the spatial wavefunction. While this may not be intuitively satisfying, it is a fundamental postulate of quantum mechanics.
  • #1
Avatrin
245
6
I know that the formula for the expectation value is:
<Q(x,p)> = ∫ψ*Q(x,(h/i)d/dx)ψ dx

For instance, the expectation value for momentum is.
-ih∫ψ*(dψ/dx)dx

But, why? How is it derived?
 
Physics news on Phys.org
  • #2
First I want to make clear that Quantum Mechanics cannot be derived from classical mechanics (including statistical mechanics and relativity); it requires some fundamentally new postulates/axioms. Different books start out with a different set of postulates and axioms, so it's conceivable that some books do not derive that formula for the expectation value, but rather state it as a postulate. For a thorough derivation of that formula, I'd refer you to a more mathematical introductory text on QM, such as Shankar's Principles of Quantum Mechanics. Granted this won't be very intuitively satisfying.

However, I can give you a non-rigorous analogy that might help you understand.

In statistics, if P(x) is the probability distribution function for the result of a measurement of some random variable x, then P(a) is the probability that a measurement of x will result in x=a. Certainly since a must be one of x's possible values,
[tex]\int_{all \ x}P(x)dx=1[/tex] This is the normalization condition.

Now for some function f(x), the expectation value of f(x), or the average of f(x), written <f(x)>, is given by: [tex]<f(x)>=\int_{all \ x}f(x)P(x)dx[/tex] This is basically a "weighted average." [From here on out, just assume any integral is taken over the entire range of possible values.]In quantum mechanics, a system's state is a ket |ψ> in a Hilbert space, and Hilbert spaces come with the L2 norm. The adjoint of |ψ> in Hilbert space is <ψ|. In math we don't use bra-ket notation, we'd use something like ψ instead of |ψ> and ψ* instead of <ψ|. The normalization condition would be 1=<ψ|ψ> and the inner product between states |χ> and |ψ> is given by <χ|ψ>; these are just fancy ways of writing a basis-free (abstract) L2 norm. (Note: I'm ignoring spin.)

Physicists often talk about the "Statistical interpretation" of |ψ>; we like to think of it as something similar to (the complex square root of) a probability distribution. [Here's where you'd like to check out Shankar for the in-depth treatment of the representation of observables by hermitian operators, and their associated eigenkets providing an orthonormal basis for the Hilbert space.] Projecting |ψ> into the position basis {|x> | x is an element of ℝ3},

∫|x><x|ψ> dx = ∫ψ(x)|x>dx

gives the normal spatial wavefunction ψ(x). Notice that (ψ(x)|x>)* = <x|ψ*(x)

The normalization condition can be written
1=∫<x|ψ*(x)ψ(x)|x>dx=∫ψ*(x)ψ(x)<x|x>dx=∫ψ*(x)ψ(x) dx
So now we are seeing similarities between P(x) and ψ*(x)ψ(x).Remember that in statistics, <f(x)> = ∫f(x)P(x)dx
We might naively assume that for some observable Q, the expectation value for Q would be
<Q> = ∫Qψ*(x)ψ(x)dx, but this is wrong. First of all, observables in quantum mechanics are represented by operators, which act on kets rather than on a scalar number like ψ*(x)ψ(x). To be technically more accurate, Q exists abstractly and acts on abstract kets in Hilbert space, so for Q to act on a spatial wavefunction, we would have to express it in the position basis. Going back to the abstract notation (no basis),
Q<ψ|ψ> is basically nonsense, since Q<ψ|ψ> = Q

The leap we have to make is realizing the correct equation must be
<Q> = <ψ|Q|ψ> = ∫ψ*(x)Qxψ(x)dx, where Qx is Q's expression in the position basis.
It's very similar to standard statistics, but because quantum physics represents observables with operators, we need to use the "sandwich" method. However, nothing I've said rigorously justifies the last leap here, so my argument is intended as an analogy rather than a derivation.

Edit: I realize I was a little sloppy in one of my equations (but of course the argument still is fine). Challenge to anyone checking my argument: can you find the sloppy equation? (And fix it?)
 
Last edited:
  • Like
Likes TEFLing

Related to Expectation Values: Formula & Derivation

1. What is an expectation value?

An expectation value is a mathematical concept used in quantum mechanics to predict the average result of a measurement of a physical quantity in a quantum system. It represents the probability-weighted average of all possible outcomes of a measurement.

2. How is the expectation value calculated?

The expectation value is calculated using the formula E = Σxp(x), where x represents the possible outcomes of the measurement and p(x) represents the probability of each outcome occurring. This formula can be derived using the principles of quantum mechanics.

3. What does the expectation value tell us about a quantum system?

The expectation value provides a prediction of the average result of a measurement in a quantum system. It can also provide information about the quantum state and the uncertainty in the measurement result.

4. Can the expectation value be negative?

Yes, the expectation value can be negative if the possible outcomes of the measurement have both positive and negative values and the probabilities of each outcome are not symmetrical. However, in most cases, the expectation value is a positive or zero value.

5. What is the relation between expectation values and uncertainty?

The expectation value can provide information about the uncertainty in a measurement. The uncertainty is represented by the standard deviation, which is the square root of the variance. The variance is calculated using the formula Var(x) = Σ(x-E)^2p(x), where x is the outcome of the measurement and E is the expectation value. A smaller standard deviation indicates a more precise measurement with less uncertainty.

Similar threads

Replies
5
Views
1K
  • Quantum Physics
Replies
3
Views
1K
Replies
14
Views
936
Replies
3
Views
1K
Replies
2
Views
1K
Replies
8
Views
1K
Replies
4
Views
1K
  • Quantum Physics
Replies
6
Views
4K
  • Quantum Physics
Replies
3
Views
1K
Replies
3
Views
2K
Back
Top