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Flashcards in Orthonormal Functions or States Deck (25)
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1
Q

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

A

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

2
Q

What is a good example to show orthogonality?

A

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

3
Q

What is dirac notation?

A

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

4
Q

What are the different components of dirac notation called?

A

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

5
Q

Outline the first postulate.

A

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

6
Q

Outline the second postulate.

A

Observables operators. Eigenvalue equation: QΨ = qΨ

7
Q

Outline the third postulate.

A

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

8
Q

Outline the fourth postulate.

A

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

9
Q

Outline the fifth postulate.

A

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

10
Q

How can we work out which operators match which observables?

A

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.

11
Q

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

A

p(hat) = -iћ∇ (3D) or -iћd/dx (1D)

12
Q

What does the operator V(hat) mean?

A

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

13
Q

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

A

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

14
Q

What is meant by the “overlap”?

A

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

15
Q

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

A

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

16
Q

How do you work out ci?

A

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

17
Q

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

A

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

18
Q

How do we separate the variables for the TDSE?

A

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

19
Q

what is the difference between the TDSE and the TISE?

A

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

20
Q

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

A

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

21
Q

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

A

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

22
Q

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

A

u(x) = u0*exp(ikx)

23
Q

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

A

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

24
Q

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

A

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

25
Q

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

A

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