Orthonormal Functions or States Flashcards
How do you determine if 2 wave functions Ψi and Ψj are orthonormal?
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
What is a good example to show orthogonality?
integral over everywhere of cos(x)*cos(2x) dx = 0, so these two functions are orthogonal.
What is dirac notation?
A compact form which, for functions maps integrals like integral over everywhere of Ψi* *Ψj dτ =
What are the different components of dirac notation called?
< | is a “bra”, | > is a “ket”, < | > is an inner product
Outline the first postulate.
Ψ tells us everything we can know about a system, |Ψ|^2 is a probability.
Outline the second postulate.
Observables operators. Eigenvalue equation: QΨ = qΨ
Outline the third postulate.
Probability of getting qi is |ci|^2, where Ψ = sum over n of cn*Ψn
Outline the fourth postulate.
Lots of measurements with same initial Ψ give us a sensible average for q, the expectation value:
Outline the fifth postulate.
Time dependence of Ψ(r,t) when we don’t make a measurement is governed by TDSE.
How can we work out which operators match which observables?
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.
What do we actually use for the equation for momentum in 3D or 1D?
p(hat) = -iћ∇ (3D) or -iћd/dx (1D)
What does the operator V(hat) mean?
Simply means multiply by V. So x(hat) would mean multiply by x.
What does the angular momentum operator L(z)(hat) equal?
L(z)(hat) = x(hat)p(y)(hat)-y(hat)p(x)(hat) = -iћ(xd/dy - y*d/dx)
What is meant by the “overlap”?
When we measure an eigenvalue q associated with operator Q(hat), the probability of getting a particular qi is the overlap.
What is P(measure of qi and Ψ collapses to Ψi) equal to?
P = |ci|^2, ci are given by overlap integrals.
How do you work out ci?
Overlap integrals: = integral over everywhere of Ψi* *Ψ dτ
What do we assume V(r) is a function of in the TDSE and why?
Assume is not a function of time, as it is easy to separate the variables.
How do we separate the variables for the TDSE?
Ψ(r, t) = u(r)f(t), so sub this in and divide through by u(r)f(t)
what is the difference between the TDSE and the TISE?
TDSE is the time evolution for Ψ, whereas the TISE is an eigenvalue equation.
What can we try for the solution of f(t)?
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
If we want to measure the total energy, what must we operate on?
Operate on Ψ with H to get an eigenvalue Ei, and Ψ collapses to Ψi.
FOr u(x), what can we try for a solution?
u(x) = u0*exp(ikx)
How do we find p(x) using u(x)?
-iћdu/dx = p(x)u, so differentiate u(x) and sub in, then rearrange
What can we show for real k after finding p(x)?
That integral over R of u(k’)* * u(k) dx = <u> = 𝛿(k-k')</u>
What is the dirac delta “spike” function defined as?
Defined to be zero everywhere except one point and defined to integrate to infinity.
