Section 3 Flashcards
Expectation values
<f> = (-∞ ∫ ∞) dx Ψ*(x)f(x)Ψ(x)
</f>
In QM, f(x) will often be
non-commuting operators
Canonical operators
Position : x(hat) = x, r(hat v) = r(v)
Momentum: p(hat) = -iℏ ∂/∂x,
p(hat v) = -iℏ∇
Energy: E(hat) = iℏ ∂/∂t
the operator relation p(hat)φ = pφ is an example of an
eigenvalue equation
Hermitian operator
An operator satisfying [A(hat)φ]* = φ*A(hat)
QM only permits
Hermitian operators => real eigenvalues
Hilbert Space
Every wave-function is a state vector |Ψ> in an infinite-dimensional vector space
The overlap integral
<Ψ|Ψ’> = (-∞ ∫ ∞) dx Ψ*(x)Ψ’(x)
The normalised eigenstate/eigenfunctions φi of a Hermitian operator form
a complete, orthonormal basis: ideal building blocks for quantum states
States built from non-degenerate eigenstates do not have
definite quantum numbers
Inner product of a wave-function with itself (derivation)
<Ψ|Ψ> = < Σi ciφi |Σj cjφj >
= Σi,j ci*cj <φi|φj>
=> Σi |ci|^2 = 1
The probability of a composite state being found in its i eigenstate component is
Pi = |ci|^2
Expectation value
is equal to the probability-weighted average of eigenvalues
What happens in repeated measurements of a single system
measurement by A(hat) collapses the wave-function Ψ -> φi
and returns quantum number ai, with probability Pi = |ci|^2
repeated measurements return ai with
100% probability
