Orthonormal Functions or States

Question 1

How do you determine if 2 wave functions Ψi and Ψj are orthonormal?

Answer 1

If the integral over everywhere of Ψi* *Ψj dτ = 𝛿ij, where τ is the element of space (dxdydz), and 𝛿ij is the Kronecker delta. 𝛿ij = 1 -> ΨiΨj normalised, 𝛿ij=0 -> ΨiΨj orthonormal

Question 2

What is a good example to show orthogonality?

Answer 2

integral over everywhere of cos(x)*cos(2x) dx = 0, so these two functions are orthogonal.

Question 3

What is dirac notation?

Answer 3

A compact form which, for functions maps integrals like integral over everywhere of Ψi* *Ψj dτ =

Question 4

What are the different components of dirac notation called?

Answer 4

< | is a “bra”, | > is a “ket”, < | > is an inner product

Question 5

Outline the first postulate.

Answer 5

Ψ tells us everything we can know about a system, |Ψ|^2 is a probability.

Question 6

Outline the second postulate.

Answer 6

Observables operators. Eigenvalue equation: QΨ = qΨ

Question 7

Outline the third postulate.

Answer 7

Probability of getting qi is |ci|^2, where Ψ = sum over n of cn*Ψn

Question 8

Outline the fourth postulate.

Answer 8

Lots of measurements with same initial Ψ give us a sensible average for q, the expectation value:

Question 9

Outline the fifth postulate.

Answer 9

Time dependence of Ψ(r,t) when we don’t make a measurement is governed by TDSE.

Question 10

How can we work out which operators match which observables?

Answer 10

Use classical case: E = p^2/2m + V. In TISE: -ћ^2/2m ∇^2 Ψ + VΨ = EΨ. Compare these to show which part equals which.

Question 11

What do we actually use for the equation for momentum in 3D or 1D?

Answer 11

p(hat) = -iћ∇ (3D) or -iћd/dx (1D)

Question 12

What does the operator V(hat) mean?

Answer 12

Simply means multiply by V. So x(hat) would mean multiply by x.

Question 13

What does the angular momentum operator L(z)(hat) equal?

Answer 13

L(z)(hat) = x(hat)p(y)(hat)-y(hat)p(x)(hat) = -iћ(xd/dy - y*d/dx)

Question 14

What is meant by the “overlap”?

Answer 14

When we measure an eigenvalue q associated with operator Q(hat), the probability of getting a particular qi is the overlap.

Question 15

What is P(measure of qi and Ψ collapses to Ψi) equal to?

Answer 15

P = |ci|^2, ci are given by overlap integrals.

Question 16

How do you work out ci?

Answer 16

Overlap integrals: = integral over everywhere of Ψi* *Ψ dτ

Question 17

What do we assume V(r) is a function of in the TDSE and why?

Answer 17

Assume is not a function of time, as it is easy to separate the variables.

Question 18

How do we separate the variables for the TDSE?

Answer 18

Ψ(r, t) = u(r)f(t), so sub this in and divide through by u(r)f(t)

Question 19

what is the difference between the TDSE and the TISE?

Answer 19

TDSE is the time evolution for Ψ, whereas the TISE is an eigenvalue equation.

Question 20

What can we try for the solution of f(t)?

Answer 20

f(t) = exp(iwt) -> df/dt = iw(exp(iwt), therefore iћiwexp(iwt) = E*exp(iwt) -> E = -ћw, so f(t) = exp(-iE(t)/ћ), a complex number on the unit circle

Question 21

If we want to measure the total energy, what must we operate on?

Answer 21

Operate on Ψ with H to get an eigenvalue Ei, and Ψ collapses to Ψi.

Question 22

FOr u(x), what can we try for a solution?

Answer 22

u(x) = u0*exp(ikx)

Question 23

How do we find p(x) using u(x)?

Answer 23

-iћdu/dx = p(x)u, so differentiate u(x) and sub in, then rearrange

Question 24

What can we show for real k after finding p(x)?

Answer 24

That integral over R of u(k’)* * u(k) dx = <u> = 𝛿(k-k')</u>

Question 25

What is the dirac delta “spike” function defined as?

Answer 25

Defined to be zero everywhere except one point and defined to integrate to infinity.