operators Flashcards

1
Q

whats the Ĥ

A

Ĥ = -ℏ²/2m * ∂²/∂x² + V(x)

the Hamiltonian operator (energy)

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2
Q

how do you calculate the expectation value of something?

A

<a> = ∫Ψ*ÂΨ dx
over all of space</a>

A is anything and  is its operator
so E and Ĥ
or x and X̂

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3
Q

What is the expectation value of energy E for a superimposed solution to the TDSE

A

superimposed: Ψ = A₁ψ₁(x)e^-iE₁t/ℏ + A₂ψ₂(x)e^-iE₂t/ℏ

<E> = E₁|A₁|² + E₂|A₂|²
</E>

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4
Q

momentum operator

A

p̂ = -iℏ*∂/∂x

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5
Q

What’s a hermitian operator

A

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)

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6
Q

whats a hermitian conjugate

A

symbol: A† (should have a hat on it)

∫ψ₁A†ψ₂ dx = ∫(Âψ₁)ψ₂ dx
important: (Âψ₁)* take the conjugate of Âψ₁ not just ψ₁

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7
Q

evaluate [Â, B̂] and explain its meaning

A

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

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8
Q

what does the result of a commutator tell you?

A

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

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9
Q

What is the generalised uncertainty principle?

A

ΔA * ΔB = |½i * <[Â, B̂]> |

if [Â, B̂] is constant (like for A = x and B = p) then
<[Â, B̂]> is just that constant

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10
Q
A
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