The 6 postulates of quantum mechanics
The 6 postulates of quantum mechanics
(i) Postulate of quantum states: All quantum states are elements of a
Hilbert space.
“ket” state: |ψi; “bra” state: hψ| = |ψi†
The 6 postulates of quantum mechanics
(i) Postulate of quantum states: All quantum states are elements of a
Hilbert space.
“ket” state: |ψi; “bra” state: hψ| = |ψi†
(ii) Observables postulate: All physical parameters (observables) have a
Hermitian operator associated with them. e.g. x → x̂, p → p̂, E → H
The 6 postulates of quantum mechanics
(i) Postulate of quantum states: All quantum states are elements of a
Hilbert space.
“ket” state: |ψi; “bra” state: hψ| = |ψi†
(ii) Observables postulate: All physical parameters (observables) have a
Hermitian operator associated with them. e.g. x → x̂, p → p̂, E → H
(iii) Expectation values postulate:
The average value for measurement of observable O in state |ψi is
R
hOiψ = ψ ∗ Ôψ dx = hψ|Ôψi = hψ|Ô|ψi
The 6 postulates of quantum mechanics
(i) Postulate of quantum states: All quantum states are elements of a
Hilbert space.
“ket” state: |ψi; “bra” state: hψ| = |ψi†
(ii) Observables postulate: All physical parameters (observables) have a
Hermitian operator associated with them. e.g. x → x̂, p → p̂, E → H
(iii) Expectation values postulate:
The average value for measurement of observable O in state |ψi is
R
hOiψ = ψ ∗ Ôψ dx = hψ|Ôψi = hψ|Ô|ψi
(iv) Completeness postulate: The eigenvalues of Ô are called its spectrum. If
the spectrum is discrete then any ψ(x) in Hilbert space can be written as
P
P
P
ψ(x) = n cn ψn (x),
|ψi = | n cn ψn i = n cn |ψn i
where the |ψn i are the eigenstates for O.
The 6 postulates of quantum mechanics
(i) Postulate of quantum states: All quantum states are elements of a
Hilbert space.
“ket” state: |ψi; “bra” state: hψ| = |ψi†
(ii) Observables postulate: All physical parameters (observables) have a
Hermitian operator associated with them. e.g. x → x̂, p → p̂, E → H
(iii) Expectation values postulate:
The average value for measurement of observable O in state |ψi is
R
hOiψ = ψ ∗ Ôψ dx = hψ|Ôψi = hψ|Ô|ψi
(iv) Completeness postulate: The eigenvalues of Ô are called its spectrum. If
the spectrum is discrete then any ψ(x) in Hilbert space can be written as
P
P
P
ψ(x) = n cn ψn (x),
|ψi = | n cn ψn i = n cn |ψn i
where the |ψn i are the eigenstates for O.
(v) Postulate for time evolution: Time evolution is determined by H,
formally
∂
i~ ∂t
|Ψ, ti = H|Ψ, ti =⇒ |Ψ, ti = e −itH/~ |Ψ, 0i
The 6 postulates of quantum mechanics
(i) Postulate of quantum states: All quantum states are elements of a
Hilbert space.
“ket” state: |ψi; “bra” state: hψ| = |ψi†
(ii) Observables postulate: All physical parameters (observables) have a
Hermitian operator associated with them. e.g. x → x̂, p → p̂, E → H
(iii) Expectation values postulate:
The average value for measurement of observable O in state |ψi is
R
hOiψ = ψ ∗ Ôψ dx = hψ|Ôψi = hψ|Ô|ψi
(iv) Completeness postulate: The eigenvalues of Ô are called its spectrum. If
the spectrum is discrete then any ψ(x) in Hilbert space can be written as
P
P
P
ψ(x) = n cn ψn (x),
|ψi = | n cn ψn i = n cn |ψn i
where the |ψn i are the eigenstates for O.
(v) Postulate for time evolution: Time evolution is determined by H,
formally
∂
i~ ∂t
|Ψ, ti = H|Ψ, ti =⇒ |Ψ, ti = e −itH/~ |Ψ, 0i
(vi) Measurement postulate: A measurement of O results in one of its
eigenvalues, αn . Probability for measuring αn is |cn |2 (sum for degenerate
states). Immediately after meas., state “collapses” to one of Ô’s
eigenstates with the measured eigenvalue. collapse:
meas.
|ψi −→
|ψn i