Expectation value of the square of the observable

In summary, The expectation value of the square of an observable is a mathematical concept used in quantum mechanics to calculate the average value of the square of a particular observable. This is done by taking the square of the wave function and integrating it over all possible values of the observable. This value is significant as it represents the average value over an infinite number of measurements and can also provide information about the uncertainty or spread of the observable's values. It can be negative and is related to the uncertainty principle in providing information about the uncertainty of an observable's values.
  • #1
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Homework Statement


I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square?

Homework Equations


[tex]\langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx[/tex]
[tex]\langle Q^2 \rangle = \int_{-\infty}^{\infty} \Psi^* (\hat{Q} \Psi)^2 \; dx \; ?[/tex]
 
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  • #2
No, the square is defined by double application of the operator

[tex] A^2 \psi =A\left(A \psi \right) \ , \forall \psi\in D(A) \ \mbox{and} \ {} A\psi \in D(A)[/tex]
 
  • #3
I tried double application after I posted and got more elegant answers. Thanks dextercioby.
 

1. What is the expectation value of the square of an observable?

The expectation value of the square of an observable is a mathematical concept used in quantum mechanics to calculate the average value of the square of a particular observable, such as position, momentum, or energy.

2. How is the expectation value of the square of an observable calculated?

The calculation of the expectation value of the square of an observable involves taking the square of the wave function of the system and integrating it over all possible values of the observable. This integral is then divided by the total probability of the system to obtain the expectation value.

3. What is the significance of the expectation value of the square of an observable?

The expectation value of the square of an observable is significant because it represents the average value of the observable over an infinite number of measurements. It can also provide information about the uncertainty or spread of the observable's values.

4. Can the expectation value of the square of an observable be negative?

Yes, the expectation value of the square of an observable can be negative. This is because it is calculated by taking the square of the wave function, which can have both positive and negative values, and integrating over all possible values of the observable.

5. How does the expectation value of the square of an observable relate to the uncertainty principle?

The expectation value of the square of an observable is related to the uncertainty principle in that it can be used to calculate the uncertainty or spread of the observable's values. According to the uncertainty principle, it is impossible to know both the exact value of an observable and its momentum at the same time, and the expectation value of the square of the observable can provide information about this uncertainty.

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