Position and Hermitian Operators

Question 1

What is a position operator?

Answer 1

For example r(hat) -> r (multiply by r)

Question 2

What is a position operator?

Answer 2

For example r(hat) -> r (multiply by r)

Question 3

What is the equation relating position operators, eigenvalues and eigenfunctions?

Answer 3

x(hat) u(x) = x0u(x), where x0 is the eigenvalue and u(x) is the eigenfunction.

Question 4

What function for u(x) has the property where multiplying by x is the same as multiplying by a constant x0?

Answer 4

u(x) = 𝛿(x-x0)

Question 5

What is the equation for normalised u(x)?

Answer 5

Integral from -inf to inf of |u(x)|^2 = 1

Question 6

What is the equation for the operator complex conjugate for Q(hat) and f?

Answer 6

Q(hat)* f = (Q(hat)f)

Question 7

When is Q(hat) Hermitian?

Answer 7

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

Question 8

What are the two key properties of Hermitian operators?

Answer 8

Eigenvalues are real and eigenfunctions form an orthogonal set.

Question 9

For the eigen value equation Q(hat)Ψn = qnΨn, how do we apply

Answer 9

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

Question 10

What do we find

Question 11

What do we find

Question 12

What do we finally find after finding

Answer 12

Can rearrange ant take out qn and qn* from sides, and the integrals equal 1, so qn = qn* (eigenvalues are real)

Question 13

What is the expectation value?

Answer 13

The “average” results of many measurements with the same starting quantum state.

Question 14

For an observale α, how can we represent it by the Hermitian operator Q(hat)?

Answer 14

= integral over everywhere of Ψ* Q(hat)Ψ dτ = -> often stated as a postulate

Question 15

What does the observable α represent in this case?

Answer 15

is the mean value of the observable α after lots of repeated measurement with the same initial Ψ.

Question 16

What is the equation for the standard deviation of a distribution j?

Answer 16

σ = sqrt( - ^2)

Question 17

How can we expand Ψ and Ψ*?

Answer 17

With linear combinations of Ψn: Ψ = sum over n of cnΨn, Ψ = (sum over m of cmΨm)

Question 18

How can we expand Ψ and Ψ*?

Answer 18

With linear combinations of Ψn: Ψ = sum over n of cnΨn, Ψ = (sum over m of cmΨm)

Question 19

What is the equation relating position operators, eigenvalues and eigenfunctions?

Answer 19

x(hat) u(x) = x0u(x), where x0 is the eigenvalue and u(x) is the eigenfunction.

Question 20

What function for u(x) has the property where multiplying by x is the same as multiplying by a constant x0?

Answer 20

u(x) = 𝛿(x-x0)

Question 21

What is the equation for normalised u(x)?

Answer 21

Integral from -inf to inf of |u(x)|^2 = 1

Question 22

What is the equation for the operator complex conjugate for Q(hat) and f?

Answer 22

Q(hat)* f = (Q(hat)f)

Question 23

When is Q(hat) Hermitian?

Answer 23

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

Question 24

What are the two key properties of Hermitian operators?

Answer 24

Eigenvalues are real and eigenfunctions form an orthogonal set.

Question 25

For the eigen value equation Q(hat)Ψn = qnΨn, how do we apply

Answer 25

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

Question 26

What do we do after finding

Answer 26

Use operator complex conjugate equation in complex conjugate of initial eigenvalue equation.

Question 27

What do we find

Question 28

What do we finally find after finding

Answer 28

Can rearrange ant take out qn and qn* from sides, and the integrals equal 1, so qn = qn* (eigenvalues are real)

Question 29

What is the expectation value?

Answer 29

The “average” results of many measurements with the same starting quantum state.

Question 30

What is the equation for the expectation value of Q(hat)?

Answer 30

= integral over everywhere of Ψ* Q(hat)Ψ dτ =

Question 31

For an observale α, how can we represent it by the Hermitian operator Q(hat)?

Answer 31

= integral over everywhere of Ψ* Q(hat)Ψ dτ = -> often stated as a postulate

Question 32

What does the observable α represent in this case?

Answer 32

is the mean value of the observable α after lots of repeated measurement with the same initial Ψ.

Question 33

What is the equation for the standard deviation of a distribution j?

Answer 33

σ = sqrt( - ^2)

Question 34

How can we expand Ψ and Ψ*?

Answer 34

With linear combinations of Ψn: Ψ = sum over n of cnΨn, Ψ = (sum over m of cmΨm)

Question 35

After rearranging the integral for , what do we find?

Answer 35

I = sum over n of |cn|^2 *qn, where I is the expectation value of Q(hat)

Question 36

What is the commutator of 2 operators?

Answer 36

[Q(hat), R(hat)] = Q)hat)R(hat) - R(hat)Q(hat)

Question 37

What would be the first step to calculate [x(hat), p(x)(hat)]?

Answer 37

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

Question 38

What do we find [x(hat), p(x)(hat)] is equal to?

Answer 38

[x(hat), p(x)(hat)] = iћ =/ 0

Question 39

What does it mean if [Q(hat), R(hat)] = 0?

Answer 39

Means that the two operators commute and are compatible functions. If =/ 0, they are incompatible functions.

Question 40

What is the equation for the generalised HUP?

Answer 40

ΔαΔβ >= 1/2i *||

Question 41

What is the variances version of the generalised HUP?

Answer 41

σ(Q)^2 *σ(R)^2 >= (1/2i *)^2

Question 42

What must compatible operators share?

Answer 42

A common set of eigenfunctions: Q(hat)Ψn = qnΨn, R(hat)Ψn = rnΨn

Question 43

How do we find the equation for [Q(hat), R(hat)] using the common set of eigenfuctions?

Answer 43

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

Question 44

What is the Hamiltonian operator H(hat) equal to?

Answer 44

H(hat) = -ћ^2/2m *∇^2 + V

Question 45

How do we work out the variance of H1: σ(H)^2?

Answer 45

= 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

Question 46

What does the result for σ(H)^2 mean?

Answer 46

Means that Ψ(r) is a stationary state and V(r) is not time dependent -> total energy E is fixed.