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