Equations Flashcards
Hamiltonian
H=p^2/2m+V(x)=-ℏ²/2m*∆²+V(X)
Schrodinger Equation for Infinite potential well
Hψ=-ℏ²/2m*d²/dx²Ψ=ΕΨ
General solution to wavefunction for infinite potential well
Ψ(x) = Asin(root(2mE/ℏ²)x²)+Bcos(root(2mE/ℏ²)x²)
Energy of infinite potential well
E=n²ℏ²π²/2ma²
Probability of finding particle at x
P(x) = |Ψ(x)|²
normalisation
<Ψ|Ψ>= ∫p(x)dx = ∫Ψ*(x)Ψ(x)dx =1
orthoganality
∫φ*ₙ(x)φₘ(x)dx = δₙₘ
δₙₘ=1 if n=m
δₙₘ= 0 if n≠m
uncertainty principle
ΔxΔp≥ℏ/2
Time dependent schrodinger equation
ĤΨ=iℏd/dtΨ (curly d)
Inner product of 2 states
∫Φ*(x)Ψ(x)dx = <φ|Ψ>
Time dependent Ψ
Ψ(x,t) = ΣcₙΨₙ(x)e^-iEt/ℏ
compact notation for energy eigenstates
Ĥ|Ψ>=E|Ψ>
<Φ|Ο|Ψ>
<Φ|Ο|Ψ> = ∫Φ*ΟΨdx
<Φ|Ο|Ψ>^† (Hermitian Conjugate)
<Φ|Ο|Ψ>^† = <Φ|Ο|Ψ>* = ∫ΦΟΨdx
Expectation value of operator
<Ο> = <Ψ|Ο|Ψ>/<Ψ|Ψ>
</Ο>
Resolution of identity
1 = Σ|aₙ><aₙ|
Dirac notation for finding expectation value of operator
<a> = <Ψ|A|Ψ> = Σ<aₙ|cₙAΣcₘ|aₘ> = Σcₙcₘ<aₙ|A|aₘ> (A|aₘ> = aₘ|aₘ>) = Σcₙcₘaₘ<aₙ|aₘ> = Σcₙcₘaₘδₙₘ = Σ|cₙ|^2aₙ</a>
When do 2 observables share eigenstates |aₙ>=|bₙ>
Eigenstates are shared when two observables are compatible. Compatible states always commute so eigenstates are also shared for commuting observables.
Show 2 compatible eigenstates commute
BA|aₙ> = Baₙ|aₙ>=aₙbₙ|aₙ>
AB|aₙ> = Abₙ|aₙ>=aₙbₙ|bₙ>
So (AB-BA)|aₙ> = 0
Commutator
[A, B] = AB-BA
Show 2 observables are compatible if they commute
[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
What is value of commutator of ladder operators â and â^†
[â, â†] = 1
affect of down ladder operator â on |Ψ₀>
â|Ψ₀> = 0
Hamiltonian for harmonic oscillator
Ĥ = P²/2m+1/2mωx^² (x and P operator)
