Understanding Expectation Value in Quantum Mechanics: A Closer Look

[GAGS]

We all know the concept of expectation value,it is the average of all possible outcomes of an experiment. Mathematically average of x is written as (Σnkxk / Σnk ). Quantum-mechanically nk is represented by probability density(P), where P = ∫Ψ*Ψ d3r,
then <r> = ∫ r P(r) d3r -----------(1)
or <r> = ∫ Ψ*(r) r Ψ(r) d3r--------(2) (normalisation condition)
but when we consider expectation value of Energy operator (= ih∂/∂t) the two equations ,to me, not give the same results. Can anybody solve what’s that dilemma and where I am wrong

 

[pmb_phy]

We all know the concept of expectation value,it is the average of all possible outcomes of an experiment. Mathematically average of x is written as (Σnkxk / Σnk ). Quantum-mechanically nk is represented by probability density(P), where P = ∫Ψ*Ψ d3r,
then <r> = ∫ r P(r) d3r -----------(1)
or <r> = ∫ Ψ*(r) r Ψ(r) d3r--------(2) (normalisation condition)
but when we consider expectation value of Energy operator (= ih∂/∂t) the two equations ,to me, not give the same results. Can anybody solve what’s that dilemma and where I am wrong

The expectation of an operator H is defined as <$\Psi$|H|$\Psi$>. The form (1) above holds only under special circumstances.

Note: H = ih∂/∂t is the energy operator only when expressed in the position representation.

Pete

 

[denian]

Thank you for your explanation of expectation value in quantum mechanics. I understand the importance of understanding this concept in order to accurately analyze and interpret experimental results.

In regards to your question about the discrepancy between equations (1) and (2) when considering the expectation value of the energy operator, it is important to note that equation (1) is the general expression for expectation value in quantum mechanics, while equation (2) is specific to the position operator.

The expectation value of the energy operator is given by the equation <E> = ∫Ψ* (ih∂/∂t) Ψ d3r. This equation takes into account the time-dependence of the wavefunction, which is not present in equation (2). This is why the two equations do not give the same results.

Additionally, the energy operator is a Hermitian operator, meaning that its expectation value will always be a real number. This may also contribute to the difference in results between the two equations.

I hope this helps clarify the dilemma you were experiencing. It is important to carefully consider the specific operators and equations being used when calculating expectation values in quantum mechanics.