QM fundamentals

Question 1

Hermitian operator

Answer 1

Adaggar = A

Can move A over WF w/o gaining -

Question 2

evals of Hermitian operator

Answer 2

Always real

Question 3

evects/efuncs of Hermitian operator

Answer 3

Form a complete orthogonal basis for H (L^2(R)) square integrable functions

Question 4

Observables represented by

Answer 4

Hermitian operators

Question 5

Position operator

Answer 5

$x\hat \psi = x \psi$

Question 6

Momentum operator

Answer 6

$p \hat \psi = \frac{h}{i} \frac{d}{dx} \psi(x)$

Question 7

Fundamental commutation relation in QM

Answer 7

x\hat and p\hat do NOT commute!

Question 8

what is [x\hat,p\hat] =

Answer 8

i h

Question 9

are x\hat and p\hat hermitian

Answer 9

Yes

Question 10

kinetic energy operator

Answer 10

p\hat ^2 /2m
Note p\hat ^2 \psi = p\hat (\frac{h}{i} \frac{d}{dx} \psi(x))
=-h^2 \frac{d^2}{dx^2}

thus
\frac{p\hat ^2}{ 2m} \psi (x) = -\frac{h^2}{2m} \frac{d^2}{dx^2} \psi (x)

Question 11

H\hat \psi (x) =

Answer 11

(K\hat +V\hat) \psi(x)

=[ -\frac{h^2}{2m} \frac{d^2}{dx^2} \psi (x) + V(x) ] \psi (x)

Question 12

What does the hamiltonian operator represent

Answer 12

The systems total energy

SUm of the the kinetic and potantial energy operators acting of \psi (x)

Question 13

What is an operator

Answer 13

A ‘recipie’ for tunring one WF into another via a combination of differentiaiton(wrt x) and multiplication (of x and scalar constants)

Question 14

What hapeneds during a measurement of a WF

Answer 14

state of system collapses from linear combination or a single eigen state. (with with meausred eval) this step is irreversible

Question 15

state will well defiend values of O are what

Answer 15

eigen states of O\hat

Since a measurement of O will yeild (with prob = 1) the eiegen value associated wiht that estate

Question 16

eigstates of x are states which

Answer 16

have a well definied eval of x

thus they havea well definied position for which $\sigma_x = 0$