(1) At a fixed time t_{0} the state of a physical system is defined by
specifying a ket |Ψ(t_{0})> 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(t_{0})> and (a_{ }e^{iθ})|Ψ(t_{0})>.
They represent the same physical state.

But |Φ> = a_{1}e^{iθ1}|Ψ_{1}> + a_{2}e^{iθ2}|Ψ_{2}>
≠ a_{1}|Ψ_{1}> + a_{2}|Ψ_{2}>
if θ_{2} ≠ θ_{2}.

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 a_{n} (a_{α})
is given by

P(a_{n}) = ∑_{i=1}^{gn}|<u_{n}^{i}|Ψ> |^{2},
(dP(a_{α}) = |<v_{α}|Ψ> |^{2}dα),

where {|u_{n}^{i}>} (i = 1, 2, ..., g_{n}) is an
orthonormal basis in the eigensubspace ℰ_{n} associated with the eigenvalue a_{n};
(where |v_{α}> is the eigenvector
corresponding to the eigenvalue a_{α}; we assume
a_{α} is a non degenerate continuous eigenvalue
of A.)

This may be written in terms of the projector

P(a_{n}) = <Ψ|P_{n}|Ψ>, P_{n} = ∑_{i=1}^{gn}|u_{n}^{i}><u_{n}^{i}|,
(dP(a_{α}) = <Ψ|P_{α}|Ψ>, P_{α}
= |v_{α}><v_{α}|dα).

(5) If a measurement of a system in the state |Ψ> gives the result a_{n}, then the state of the system immediately after the
measurement is the normalized projection of |Ψ> onto
the eigensubspace associated with a_{n};

|Ψ> --> P_{n}|Ψ>/(<Ψ|P_{n}|Ψ>)^{½}
after measuring a_{n}.

(6) The Cartesian components of the observables **R** and
**P**
satisfy [R_{i},R_{j}] = [P_{i},P_{j}] = 0, [R_{i},P_{j}]
= ihδ_{ij}. These are called the canonical commutation relations.
**R** is the position operator and
**P** is the operator corresponding to
the **c**onjugate 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 p_{x} = ∂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 = p^{2}/(2m) + U(**r**,t).

This is an expression for the total energy
of the system.

The quantum mechanical operator is found by replacing p^{2} with P^{2}
= P_{x}^{2} + P_{y}^{2} + P_{z}^{2}
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)> = [P^{2}/(2m) + U(**R**,t)]|Ψ(t)>.

The Schroedinger equation is first order in t. Given |Ψ(t_{0})>, |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} = (<A^{2}> - <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 = A

A Hermitian operator satisfies <Ψ|A|Φ> = <Φ|A|Ψ>

An operator A is anti Hermitian if A = -A

**Unitary operators
**An operator U is unitary if UU

<UΨ|UΨ> = <Ψ|U

**The |r>
and |p> representations
**Two orthonormal, continuously labeled bases for the vector space L

{u

with Ψ(x) = ∫

and

{δ

with Ψ(x) = ∫

In three dimensions the bases are

{u_{p0}(**r**) = (2πħ)^{-3/2}exp(i(**p**_{0}/ħ)∙**r**)},

with Ψ(**r**) = ∫_{∞}^{+∞}
d^{3}p_{0} Ψ(**p**_{0})u_{p0}(**r**)
= (2πħ)^{-3/2}∫_{∞}^{+∞}
d^{3}p_{0} Ψ(**p**_{0})exp(i(**p**_{0}/ħ)∙**r**),

and

{δ_{r0}(**r**) =
δ(**r** - **r**_{0})}

with Ψ(**r**) = ∫_{∞}^{+∞}
d^{3}**r**_{0} Ψ(**r**_{0})δ_{r0}(**r**)
= ∫_{∞}^{+∞}
d^{3}**r**_{0} Ψ(**r**_{0})δ(**r**
- **r**_{0}).

Let us associate the kets |**p**_{0}> with u_{p0}(**r**)and |**r**_{0}>
with
δ_{r0}(**r**). Then
Ψ(**p**_{0})
= <**p**_{0}|Ψ> denotes the components of |Ψ>
in the {|**p**_{0}>} basis and Ψ(**r**_{0})
= <**r**_{0}|Ψ> denotes
the components of |Ψ> in the {|**r**_{0}>}
basis.**The R and P operators
**In the {|

(ħ/i)(∂/∂x). Similarly,

P

In the {|

The evolution operator

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,t