Quantum Mechanics: Questions on Stationary Particles & Wave Function

[neelakash]

I have started my course (a bit earlier than my university---that is why I am reading myself) in Quantum Physics.And I got the following queries to clarify:
Sometimes we need to accept seemingly contradictory features...I want to discuss them and clarify...

(i)Can we conceive a stationary (at rest) particle in QM?If so,do we need to associate a wave-function with it?

What I think:Qm is not worried about this.Even if it is possible,there will be no physically interesting situation...

(ii)In the expression of a(p),which is the Fourier transform of Ψ(x),time t is explicit.Yet, a(p) does not depend on time.

What I think:I do not understand why.

(iii) We know: <x>=∫(Ψ* x Ψ) dx and <p>=(ˉh/i) ∫[Ψ* (∂/∂x) Ψ] dx
where (h/2π)= ˉh

Why is Ψ* in the front place and not the Ψ?What would be the problem if the formula were:

<x>=∫(Ψ x Ψ*) dx and <p>=(ˉh/i) ∫[Ψ (∂/∂x) Ψ*] dx

What I think: I do not understand.

 

[carroza]

(i)Can we conceive a stationary (at rest) particle in QM?If so,do we need to associate a wave-function with it?

What I think:Qm is not worried about this.Even if it is possible,there will be no physically interesting situation...

Stationary, and at rest, do not mean the same in QM.

A wavefunction is stationary in QM if its probability density does not change with time. This is the case for all the eigenstates of a hamiltonian.

Classically, a particle at rest will have a definite position and a definite momentum (p=0), in a given
instant of time. This is not possible in quantum mechanics. And, as quantum mechanics is (so far)
succesful in describing nature, there are no particles strictly at rest in nature.

If you define "at rest", as something having zero momentum, one can have such a particle in QM, and it will be described by a wavefunction, given by a constant Ψ(x)=C, which will have zero momentum, but the position will be completely undetermined.


Have fun with quantum mechanics. Your other questions will be clear once that you follow your QM course.