operators Flashcards
whats the Ĥ
Ĥ = -ℏ²/2m * ∂²/∂x² + V(x)
the Hamiltonian operator (energy)
how do you calculate the expectation value of something?
<a> = ∫Ψ*ÂΨ dx
over all of space</a>
A is anything and  is its operator
so E and Ĥ
or x and X̂
What is the expectation value of energy E for a superimposed solution to the TDSE
superimposed: Ψ = A₁ψ₁(x)e^-iE₁t/ℏ + A₂ψ₂(x)e^-iE₂t/ℏ
<E> = E₁|A₁|² + E₂|A₂|²
</E>
momentum operator
p̂ = -iℏ*∂/∂x
What’s a hermitian operator
A = A†
all eigenvalues are real (most physical observables are hermitian)
all eigenfunctions with different eigenvalues are orthogonal
meaning: ∫ψ₁ψ₂ dx = 0
ψ₁ is the conjugate (* is not a multiply symbol)
whats a hermitian conjugate
symbol: A† (should have a hat on it)
∫ψ₁A†ψ₂ dx = ∫(Âψ₁)ψ₂ dx
important: (Âψ₁)* take the conjugate of Âψ₁ not just ψ₁
evaluate [Â, B̂] and explain its meaning
ÂB̂ - B̂Â
the commutator or  and B̂ is the difference between measuring B then A and measuring A then B. (for ÂB̂ B is measured first since operators act to the right)
if [Â, B̂] = 0 then A and B are compatible and share eigenfunctions
what does the result of a commutator tell you?
if [Â, B̂] = 0 then A and B are compatible and share eigenfunctions
if [Â, B̂] != 0 then A and B are incompatible. measuring one affects the other. It’s impossible to know both simultaneously
What is the generalised uncertainty principle?
ΔA * ΔB = |½i * <[Â, B̂]> |
if [Â, B̂] is constant (like for A = x and B = p) then
<[Â, B̂]> is just that constant