Equations

Question 1

Hamiltonian

Answer 1

H=p^2/2m+V(x)=-ℏ²/2m*∆²+V(X)

Question 2

Schrodinger Equation for Infinite potential well

Answer 2

Hψ=-ℏ²/2m*d²/dx²Ψ=ΕΨ

Question 3

General solution to wavefunction for infinite potential well

Answer 3

Ψ(x) = Asin(root(2mE/ℏ²)x²)+Bcos(root(2mE/ℏ²)x²)

Question 4

Energy of infinite potential well

Answer 4

E=n²ℏ²π²/2ma²

Question 5

Probability of finding particle at x

Answer 5

P(x) = |Ψ(x)|²

Question 6

normalisation

Answer 6

<Ψ|Ψ>= ∫p(x)dx = ∫Ψ*(x)Ψ(x)dx =1

Question 7

orthoganality

Answer 7

∫φ*ₙ(x)φₘ(x)dx = δₙₘ
δₙₘ=1 if n=m
δₙₘ= 0 if n≠m

Question 8

uncertainty principle

Answer 8

ΔxΔp≥ℏ/2

Question 9

Time dependent schrodinger equation

Answer 9

ĤΨ=iℏd/dtΨ (curly d)

Question 10

Inner product of 2 states

Answer 10

∫Φ*(x)Ψ(x)dx = <φ|Ψ>

Question 11

Time dependent Ψ

Answer 11

Ψ(x,t) = ΣcₙΨₙ(x)e^-iEt/ℏ

Question 12

compact notation for energy eigenstates

Answer 12

Ĥ|Ψ>=E|Ψ>

Question 13

<Φ|Ο|Ψ>

Answer 13

<Φ|Ο|Ψ> = ∫Φ*ΟΨdx

Question 14

<Φ|Ο|Ψ>^† (Hermitian Conjugate)

Answer 14

<Φ|Ο|Ψ>^† = <Φ|Ο|Ψ>* = ∫ΦΟΨdx

Question 15

Expectation value of operator

Answer 15

<Ο> = <Ψ|Ο|Ψ>/<Ψ|Ψ>
</Ο>

Question 16

Resolution of identity

Answer 16

1 = Σ|aₙ><aₙ|

Question 17

Dirac notation for finding expectation value of operator

Answer 17

<a> = <Ψ|A|Ψ> = Σ<aₙ|cₙAΣcₘ|aₘ> = Σcₙcₘ<aₙ|A|aₘ> (A|aₘ> = aₘ|aₘ>) = Σcₙcₘaₘ<aₙ|aₘ> = Σcₙcₘaₘδₙₘ = Σ|cₙ|^2aₙ</a>

Question 18

When do 2 observables share eigenstates |aₙ>=|bₙ>

Answer 18

Eigenstates are shared when two observables are compatible. Compatible states always commute so eigenstates are also shared for commuting observables.

Question 19

Show 2 compatible eigenstates commute

Answer 19

BA|aₙ> = Baₙ|aₙ>=aₙbₙ|aₙ>
AB|aₙ> = Abₙ|aₙ>=aₙbₙ|bₙ>
So (AB-BA)|aₙ> = 0

Question 20

Commutator

Answer 20

[A, B] = AB-BA

Question 21

Show 2 observables are compatible if they commute

Answer 21

[Â, B]|Ψ> = 0
so ÂB|bₙ> = BÂ|bₙ>= Âbₙ|bₙ>
so B(Â|bₙ>) = bₙ (Â|bₙ>)
so Â|bₙ ∝ |bₙ> so |bₙ> are eigenstates of Â

Question 22

What is value of commutator of ladder operators â and â^†

Answer 22

[â, â†] = 1

Question 23

affect of down ladder operator â on |Ψ₀>

Answer 23

â|Ψ₀> = 0

Question 24

Hamiltonian for harmonic oscillator

Answer 24

Ĥ = P²/2m+1/2mωx^² (x and P operator)

Question 25

Lowering operator

Answer 25

â = root(mω/2ℏ)(x+ip/mω) (x and p are operators)

Question 26

Raising operator

Answer 26

↠= root(mω/2ℏ)(x-ip/mω) (x and p are operators)

Question 27

x operator in terms of ladder operators

Answer 27

x = root(ℏ/2mω)(â+â†)

Question 28

p operator in terms of ladder operators

Answer 28

x = -iroot(mℏω/2)(â-â†)

Question 29

Is â hermitian?

Answer 29

NO! an operator is hermitian if H† = H. Here ↠≠ â therefore its not hermitian and therefore does not correspond to an observable

Question 30

what is â?

Answer 30

Lowering operator which produces an eigenstate with energy lowered by ℏω

Question 31

what is �

Answer 31

Raising operator which produces an eigenstate with energy raised by ℏω

Question 32

Hamiltonian in terms of ladder operators

Answer 32

Ĥ = ℏω/2(ââ†+â†â) = ℏω(ââ†+1/2) = ℏω(â†â-1/2)

Question 33

Angular momentum

Answer 33

L = r x p (cross product of radius and momentum)
(L=mvr)

Question 34

commutator of Lₓ and Ly

Answer 34

[Lₓ, Ly] = [ŷpz-zpy, zpₓ-xpz] =[ŷpz, zpx]+[zpy, xpz] = ŷ[pz,z]px +py[z,pz]x = iℏLz

Question 35

commutator of Ly and Lz

Answer 35

[Lx, Lz] = iℏLₓ

Question 36

commutator of Lz and Lx

Answer 36

[Lz, Lₓ] = iℏLy

Question 37

[A,BC]

Answer 37

[A,BC] = ABC-BCA=(AB-BA)C+B(AC-CA) = [A,B]C+B[A,C}

Question 38

[L², Lz]

Answer 38

[L², Lz] = 0
Allows us to use L² and Lz simultaneously and use them to label angular momentum eigenvalues

Question 39

Angular momentum raising operator

Answer 39

L₊=Lₓ+iLy
Raises eigenvalue of Lz
For S raising operator replace L with S

Question 40

Angular momentum raising operator

Answer 40

L₋=Lₓ-iLy
Raises eigenvalue of Lz
For S lowering operator replace L with S

Question 41

ml values

Answer 41

ml = -l, -l+1, …, l-1, l

Question 42

Lz|Yₗ,ₘₗ>

Answer 42

Lz|Yₗ,ₘₗ> = mlℏ|Yₗ,ₘₗ>

Question 43

L²|Yₗ,ₘₗ>

Answer 43

L²|Yₗ,ₘₗ> =l(l+1)ℏ²|Yₗ,ₘₗ>

Question 44

Magnitude of angular momentum

Answer 44

|l| =ℏ√l(l+1)

Question 45

L₊|Yₗ,ₘₗ>

Answer 45

L₊|Yₗ,ₘₗ> = Dₗ,ₘₗ|Yₗ,ₘₗ>

Dₗ,ₘₗ =ℏ√(l(l+1)-mₗ(mₗ+1))

Question 46

L₋|Yₗ,ₘₗ>

Answer 46

L₋|Yₗ,ₘₗ> = Cₗ,ₘₗ|Yₗ,ₘₗ>
Cₗ,ₘₗ =ℏ√(l(l+1)-mₗ(mₗ-1))

Question 47

S²|χ>

Answer 47

S²|χ> = s(s+1)ℏ|χ>

Question 48

Sz|χ>

Answer 48

Sz|χ> = mₛℏ|χ>

Question 49

|χ> (spin state)

Answer 49

|χ> = c₁²|↑| + c₂²|↓|

Question 50

S₊|↑|

Answer 50

S₊|↑| = 0

Question 51

S₊|↓|

Answer 51

S₊|↓| = ℏ|↑|

Question 52

S₋|↓|

Answer 52

S₋|↓| = 0

Question 53

S₋|↑| = 0

Answer 53

S₋|↑|= ℏ|↓|