(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 eiθ)|Ψ(t0)>.
They represent the same physical state.
But |Φ> = a1eiθ1|Ψ1> + a2eiθ2|Ψ2>
≠ a1|Ψ1> + a2|Ψ2>
if θ2 ≠ θ2.
Global phase factors and multipliers do not affect the physical predictions, but
relative phases and multipliers are significant.
(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]
= iħδij. 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.
(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:
r∙p --> ½(r∙p + p∙r)
--> ½(R∙P + P∙R)
(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.
Fundamental assumptions in coordinate representation
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.
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]>|.
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)-½exp(i(p0/ħ)x)},
with Ψ(x) =
∫∞+∞
dp0 Ψ(p0)up0(x)
= (2πħ)-½∫∞+∞
dp0 Ψ(p0)exp(i(p0/ħ)x).
and
{δx0(x)
=
δ(x - x0)},
with Ψ(x) =
∫∞+∞
dx0 Ψ(x0)δx0(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
{δr0(r) =
δ(r - r0)}
with Ψ(r) = ∫∞+∞
d3r0 Ψ(r0)δr0(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)).