Postulates of QM

How is the state of a quantum mechanical system described?

(1)  At a fixed time t0 the state of a physical system is defined by specifying a ket |Ψ(t0)> belonging to the space ℰ.  ℰ is a complex, separable Hilbert space, a complex linear vector space in which an inner product is defined and which possesses a countable orthonormal basis.  The vectors in such a space have the properties mathematical objects must have in order to be capable of describing a quantum system.
Consider two kets, |Ψ(t0)> and (a e)|Ψ(t0)>.  They represent the same physical state.
But |Φ> = a1eiθ11> + a2eiθ22> ≠  a11> + a22>  if θ2 ≠ θ2.
Global phase factors and multipliers do not affect the physical predictions, but relative phases and multipliers are significant.

How can we predict the results of a measurement?

(2)  Every measurable physical quantity is described by a Hermitian operator acting in ℰ.  (We want the eigenvectors of the operator to form a basis for the vector space and its eigenvalues to be real.)
(3)  The only possible results of a measurement are the eigenvalues of the Hermitian operator describing this measurement.
(4)  When a physical quantity described by the operator A is measured on a system in a normalized state |Ψ>, the probability of measuring the eigenvalue an (aα) is given by
P(an) = ∑i=1gn|<uni|Ψ> |2,    (dP(aα) = |<vα|Ψ> |2dα),
where {|uni>} (i = 1, 2, ..., gn) is an orthonormal basis in the eigensubspace ℰn associated with the eigenvalue an; (where |vα> is the eigenvector corresponding to the eigenvalue aα; we assume aα is a non degenerate continuous eigenvalue of A.)
This may be written in terms of the projector
P(an) = <Ψ|Pn|Ψ>,  Pn = ∑i=1gn|uni><uni|,    (dP(aα) = <Ψ|Pα|Ψ>,  Pα = |vα><vα|dα).  
(5)  If a measurement of a system in the state |Ψ> gives the result an, then the state of the system immediately after the measurement is the normalized projection of |Ψ> onto the eigensubspace associated with an;
|Ψ> --> Pn|Ψ>/(<Ψ|Pn|Ψ>)½ after measuring an.
(6)  The Cartesian components of the observables R and P satisfy   [Ri,Rj] = [Pi,Pj] = 0,   [Ri,Pj] = ihδij.  These are called the canonical commutation relations.  R is the position operator and P is the operator corresponding to the conjugate momentum or canonical momentum.

How do we find the operator corresponding to a physical quantity that is classically defined?

(a)  Express the physical quantity in terms of the fundamental dynamical variables r and the conjugate momenta p.  We define the conjugate momentum through px = ∂L/∂x, ... , where L = Lagrangian.
(b)  Symmetrize the expression with respect to r and p, then replace the variables r and p with the operators R and P

Example: 

rp --> ½(rp + pr) --> ½(RP + PR)  

How is the state of a quantum mechanical system evolve?

(7)  The time evolution of the state vector is governed by the Schroedinger equation,
(iħ∂/∂t)|Ψ(t)> = H(t)|Ψ(t)>, 
where H(t) is the observable associated with the total energy of the system.
For a conservative system, where all the forces can be derived by taking the gradient of a scalar potential, the classical Hamiltonian of a particle is written as 
H = T + U = p2/(2m) + U(r,t).
This is an expression for the total energy of the system.
The quantum mechanical operator is found by replacing p2 with P2 = Px2 + Py2 + Pz2 and U(r,t) with U(R,t).  No product of non-commuting operators is involved, so symmetrization is not required.  The Schroedinger equation becomes
iħ∂/∂t)|Ψ(t)> = [P2/(2m) + U(R,t)]|Ψ(t)>.
The Schroedinger equation is first order in t.  Given |Ψ(t0)>, |Ψ(t)> is uniquely determined. The quantum state evolves in a perfectly deterministic way between measurements.


Commuting observables

Two commuting observables can be measured simultaneously, i.e. the measurement of one does not cause loss of information obtained in the measurement of the other.  If we measure a complete set of commuting observables (C.S.C.O.), then the state of the system after the measurement is one element of an unique eigenbasis.  The results of the measurement specify the state completely.


Mean value and root mean square deviation of an observable

The expression for the mean value of an observable A in the normalized state |Ψ>
is <A> = <Ψ|A|Ψ>.  If |Ψ> is not normalized then  <A> = <Ψ|A|Ψ>/<Ψ|Ψ>.

The root mean square deviation ΔA characterizes the dispersion of the measurement around <A>.
ΔA = (<(A - <A>)2>)½ = (<A2> - <A>2)½.

The generalized uncertainty relation:
Let A and B be two observables (Hermitian operators).
In any state of the system   ΔA ΔB ≥ ½|<i[A,B]>|.


Important Operators

Hermitian operators
An operator A is Hermitian if A = AT.
A Hermitian operator satisfies <Ψ|A|Φ> = <Φ|A|Ψ>*.
An operator A is anti Hermitian if A = -AT.

Unitary operators
An operator U is unitary if UUT = UTU = I.  An unitary operator preserves the norm.
<UΨ|UΨ> = <Ψ|UTU|Ψ> = <Ψ|Ψ>.

The |r> and |p> representations
Two orthonormal, continuously labeled bases for the vector space L2x of square integrable functions are
{up0(x) = (2πħ)exp(i(p0/ħ)x)},
with Ψ(x) = ∫+∞ dp0 Ψ(p0)up0(x) = (2πħ)+∞ dp0 Ψ(p0)exp(i(p0/ħ)x).
and
x0(x) = δ(x - x0)},
with Ψ(x) = ∫+∞ dx0 Ψ(x0x0(x)  = ∫+∞ dx0 Ψ(x0)δ(x - x0).

In three dimensions the bases are
{up0(r) = (2πħ)-3/2exp(i(p0/ħ)∙r)},
with Ψ(r) = ∫+∞ d3p0 Ψ(p0)up0(r) = (2πħ)-3/2+∞ d3p0 Ψ(p0)exp(i(p0/ħ)∙r),
and
r0(r) = δ(r - r0)}
with Ψ(r) = ∫+∞ d3r0 Ψ(r0r0(r)  = ∫+∞ d3r0 Ψ(r0)δ(r - r0).

Let us associate the kets |p0> with up0(r)and |r0> with δr0(r).  Then Ψ(p0) = <p0|Ψ> denotes the components of |Ψ> in the {|p0>} basis and Ψ(r0) = <r0|Ψ> denotes the components of |Ψ> in the {|r0>} basis.

The R and P operators
In the {|r>} representation the operator Px coincides with the differential operator
(ħ/i)(∂/∂x).  Similarly,
Py --> (ħ/i)(∂/∂y), Pz --> (ħ/i)(∂/∂z), or P --> (ħ/i).
In the {|p>} representation X --> -(ħ/i)(∂/∂px).

The evolution operator
The evolution operator is a unitary operator defined through |Ψ(t)> = U(t,t0)|Ψ(t0)>.
U(t + dt, t) = I - (i/ħ)H(t)dt is the infinitesimal evolution operator.
If H does not explicitly depend on time, then U(t,t0) = exp(-(i/ħ)H(t - t0)).