Position and Hermitian Operators Flashcards
What is a position operator?
For example r(hat) -> r (multiply by r)
What is a position operator?
For example r(hat) -> r (multiply by r)
What is the equation relating position operators, eigenvalues and eigenfunctions?
x(hat) u(x) = x0u(x), where x0 is the eigenvalue and u(x) is the eigenfunction.
What function for u(x) has the property where multiplying by x is the same as multiplying by a constant x0?
u(x) = 𝛿(x-x0)
What is the equation for normalised u(x)?
Integral from -inf to inf of |u(x)|^2 = 1
What is the equation for the operator complex conjugate for Q(hat) and f?
Q(hat)* f = (Q(hat)f)
When is Q(hat) Hermitian?
if integral over everywhere of f* Q(hat)gdτ = integral over everywhere of gQ(hat)* f dτ = integral over everywhere of (Q(hat)f) *g dτ
What are the two key properties of Hermitian operators?
Eigenvalues are real and eigenfunctions form an orthogonal set.
For the eigen value equation Q(hat)Ψn = qnΨn, how do we apply
Times by Ψn* and integrate: integral over everywhere of Ψn* Q(hat)Ψn dτ = integral over everywhere of Ψn* qnΨn dτ = qnintegral over everywhere of Ψn *Ψn dτ
What do we find
What do we find
What do we finally find after finding
Can rearrange ant take out qn and qn* from sides, and the integrals equal 1, so qn = qn* (eigenvalues are real)
What is the expectation value?
The “average” results of many measurements with the same starting quantum state.
For an observale α, how can we represent it by the Hermitian operator Q(hat)?
= integral over everywhere of Ψ* Q(hat)Ψ dτ = -> often stated as a postulate
What does the observable α represent in this case?
is the mean value of the observable α after lots of repeated measurement with the same initial Ψ.
What is the equation for the standard deviation of a distribution j?
σ = sqrt( - ^2)
How can we expand Ψ and Ψ*?
With linear combinations of Ψn: Ψ = sum over n of cnΨn, Ψ = (sum over m of cmΨm)
How can we expand Ψ and Ψ*?
With linear combinations of Ψn: Ψ = sum over n of cnΨn, Ψ = (sum over m of cmΨm)
What is the equation relating position operators, eigenvalues and eigenfunctions?
x(hat) u(x) = x0u(x), where x0 is the eigenvalue and u(x) is the eigenfunction.
What function for u(x) has the property where multiplying by x is the same as multiplying by a constant x0?
u(x) = 𝛿(x-x0)
What is the equation for normalised u(x)?
Integral from -inf to inf of |u(x)|^2 = 1
What is the equation for the operator complex conjugate for Q(hat) and f?
Q(hat)* f = (Q(hat)f)
When is Q(hat) Hermitian?
if integral over everywhere of f* Q(hat)gdτ = integral over everywhere of gQ(hat)* f dτ = integral over everywhere of (Q(hat)f) *g dτ
What are the two key properties of Hermitian operators?
Eigenvalues are real and eigenfunctions form an orthogonal set.
For the eigen value equation Q(hat)Ψn = qnΨn, how do we apply
Times by Ψn* and integrate: integral over everywhere of Ψn* Q(hat)Ψn dτ = integral over everywhere of Ψn* qnΨn dτ = qnintegral over everywhere of Ψn *Ψn dτ
What do we do after finding
Use operator complex conjugate equation in complex conjugate of initial eigenvalue equation.
What do we find
What do we finally find after finding
Can rearrange ant take out qn and qn* from sides, and the integrals equal 1, so qn = qn* (eigenvalues are real)
What is the expectation value?
The “average” results of many measurements with the same starting quantum state.
What is the equation for the expectation value of Q(hat)?
= integral over everywhere of Ψ* Q(hat)Ψ dτ =
For an observale α, how can we represent it by the Hermitian operator Q(hat)?
= integral over everywhere of Ψ* Q(hat)Ψ dτ = -> often stated as a postulate
What does the observable α represent in this case?
is the mean value of the observable α after lots of repeated measurement with the same initial Ψ.
What is the equation for the standard deviation of a distribution j?
σ = sqrt( - ^2)
How can we expand Ψ and Ψ*?
With linear combinations of Ψn: Ψ = sum over n of cnΨn, Ψ = (sum over m of cmΨm)
After rearranging the integral for , what do we find?
I = sum over n of |cn|^2 *qn, where I is the expectation value of Q(hat)
What is the commutator of 2 operators?
[Q(hat), R(hat)] = Q)hat)R(hat) - R(hat)Q(hat)
What would be the first step to calculate [x(hat), p(x)(hat)]?
x(hat) = multiply by x, and p(x)(hat) = -iћ d/dx, so [x(hat), p(x)(hat)]Ψ(x) = x(-iћ dΨ/dx) + iћ dxΨ/dx = iћΨ
What do we find [x(hat), p(x)(hat)] is equal to?
[x(hat), p(x)(hat)] = iћ =/ 0
What does it mean if [Q(hat), R(hat)] = 0?
Means that the two operators commute and are compatible functions. If =/ 0, they are incompatible functions.
What is the equation for the generalised HUP?
ΔαΔβ >= 1/2i *||
What is the variances version of the generalised HUP?
σ(Q)^2 *σ(R)^2 >= (1/2i *)^2
What must compatible operators share?
A common set of eigenfunctions: Q(hat)Ψn = qnΨn, R(hat)Ψn = rnΨn
How do we find the equation for [Q(hat), R(hat)] using the common set of eigenfuctions?
Operate one by the other (operate R(hat) on first one), and rearrange. Find that it equals (Q(hat)R(hat)-R(hat)Q(hat))*Ψn = 0 i.e. [Q(hat), R(hat)] = 0
What is the Hamiltonian operator H(hat) equal to?
H(hat) = -ћ^2/2m *∇^2 + V
How do we work out the variance of H1: σ(H)^2?
= integral over everywhere of ΨnH(hat)Ψ dτ = Eintegral over everywhere of Ψn *Ψ dτ -> find that σ(H)^2 = E^2 - E^2 = 0
What does the result for σ(H)^2 mean?
Means that Ψ(r) is a stationary state and V(r) is not time dependent -> total energy E is fixed.
