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