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Subsections
Algebraic Quantum Physics
While the Schrödinger Equation, in principle, allows to compute all
details of the particle distribution and the exact energy terms,
having to deal with partial differential equations, boundary conditions
and normalization factors, is usually very cumbersome and often
can't be done analytically, anyway.
Just a nobody would try to develop a color TV set by solving Maxwell
equations, the discussion of complex quantum systems requires a
more abstract formalism.
The solutions from the examples in section
1.2.3.3 and 1.2.3.4 are complex functions
over the intervals
or
,
respectively.
Let's introduce the following abbreviations1.8
|
(1.24) |
or, for the case of indices
|
(1.25) |
and also introduce a scalar Product
defined as
|
(1.26) |
The scalar product
of the eigenfunctions
and from the on dimensional capacitor example
(1.2.3.3) gives
|
(1.27) |
The substitution
leads to
|
(1.28) |
So the eigenfunctions of the Hamilton operator are
orthonormal according to the scalar product (1.26)
and therefor form the base of the orthonormal vector space
consisting of all possible linear combinations
of
. This space is the Hilbert space
for this particular problem and it can be shown that the
eigenvalues of any operator describing a physical
observable form an orthogonal base.1.9
Completeness
Since the Schrödinger Equation is a linear differential equation,
any linear combination of solutions is also a solution and thus
a valid physical state.
To calculate the expectation value
of the energy for a given
state we have to solve the integral
|
(1.29) |
If is given as a sum of eigenfunctions as in
equation 1.19, integration can be avoided, as
|
(1.30) |
Since
and
,
can be expressed as a weighted sum of eigenvalues:
|
(1.31) |
Using the eigenfunctions for the one-dimensional capacitor (1.2.3.3)
the complex amplitudes for an arbitrary continuous function
over are given by
|
(1.32) |
This describes a standard sine-Fourier Transform.
The original function can be reconstructed by a composition
of eigenfunctions with the Fourier components
|
(1.33) |
As before, it can be shown that the eigenvalues of any Hamilton
operator always form a complete orthonormal base, thus
|
(1.34) |
A Hilbert space is a linear vector space over the
scalar body . Let
and
, then the following operations are
defined [23]:
|
|
|
(1.35) |
|
|
|
(1.36) |
|
|
|
(1.37) |
|
|
|
(1.38) |
The inner product
meets the following
conditions:
|
|
|
(1.39) |
|
|
|
(1.40) |
|
|
|
(1.41) |
|
|
|
(1.42) |
|
|
|
(1.43) |
As we have shown in 1.3.1.2, all valid states can
be expressed as a sum of eigenfunctions, i.e.
|
(1.44) |
If we use
as unit vectors, we can
write the bra- and ket-vectors of as infinitely
dimensional row- and column-vectors
|
(1.45) |
The time independent Schrödinger equation can then be written as
|
(1.46) |
The Hamilton Operator is the diagonal matrix
.
In the case of multiple
indices as in 1.2.3.4, a diagonalization such as e.g.
,
can be used to order the eigenfunctions.
If such an diagonalization exists for a Hilbert space , then every
linear operator of can be written in matrix form with
the matrix elements
.
|
(1.47) |
Physical Observables
As has been mentioned in 1.2.2.3, in quantum physics, a physical
observable is expressed as a linear operator (see table 1.1)
while the classical value of is the expectation value
.
Obviously, the value of an observable such as position or momentum
must be real, as a length of
meter would have no physical meaning,
so we require
.
is called adjoint operator to if
|
(1.48) |
If is given in matrix form, the is the conjugated transposition
of , i.e.
.
An operator with
is called self adjoint
or Hermitian.
All quantum observables are represented by Hermitian operators as
we can reformulate the requirement
as
or
|
(1.49) |
Measurement
In classical physics, the observables of a system such as particle location,
momentum, Energy, etc. where thought to be well defined entities which
change their values over time according to certain dynamic laws and which could
-- technical difficulties aside -- in principle be observed without
disturbing the system itself.
It is a fundamental finding of quantum physics that this is not the case.
Consider a state
which is a composition of two eigenstates
and
of the time-independent Schrödinger
equation with the assorted energy-eigenvalues and
|
(1.52) |
The expectation value of energy
, but if
we actually perform the measurement, we will measure either or
with the probabilities and .
However, if we measure the resulting state again, we will always get the same
energy as in the first measurement as the wave function has
collapsed to either or .
|
(1.53) |
The fact that
is only a statistical value, brings
up the question when it is reasonable to speak about the energy
of a state (or any other observable, for the matter) or,
with other words, whether a physical quality of a system exists
for itself or is invariably tied to the process of measuring.
The Copenhagen interpretation of quantum physics argues
that an observable only exists if the system in
question happens to be in an eigenstate of the according
operator [22].
The destructive nature of measurement raises the question
whether 2 observables and can
be measured simultaneously.
This can only be the case if the post-measurement state
is an eigenfunction of and
|
(1.54) |
Using the commutator , this is equivalent
to the condition .
If and don't commute, then the uncertainty product
(see 1.2.2.3)
.
To find a lower limit for
we introduce
the operators
and
and can express the squared uncertainty product as
|
(1.55) |
Since and are self adjoint, we express
the above as
.
Using Schwarz's Inequality
and
the fact that
we get
|
(1.56) |
Observables with a nonzero commutator of the dimension
of action (i.e. a product of energy and time) are
canonically conjugated.
If we take e.g. the location and momentum operators from
1.2.2.3, we find that
|
(1.57) |
This means that it is impossible to define the location and the
impulse for the same coordinate to arbitrary precision; it
is, however, possible the measure the location in -direction
together with the impulse in -direction.
Temporal Evolution
In 1.2.3.1 we have shown how the Schrödinger
equation can be separated if the Hamilton operator is
time independent.
If we have the initial value problem with
we can define an operator such that
|
(1.58) |
We get the operator equation
with the
solution
|
(1.59) |
is the operator of temporal evolution and satisfies
the criterion
|
(1.60) |
If
is a solution of the
time-independent Schrödinger equation, then
|
(1.61) |
is the corresponding time dependent solution
(see 1.2.3.1).
Unitary Operators
The operator of temporal evolution satisfies the condition
|
(1.62) |
Operators with
are called unitary.
Since the temporal evolution of a quantum system is described
by a unitary operator and
it follows that
the temporal behavior of a quantum system is reversible,
as long a no measurement is performed.1.10
Unitary operators can also be used to describe abstract operations
like rotations
|
(1.63) |
or the flipping of eigenstates
|
(1.64) |
without the need to specify how this transformations are actually
performed or having to deal with time-dependent Hamilton operators.
Mathematically, unitary operations can be described as
base-transformations between 2 orthonormal bases (just like
rotations in ).
Let and be Hermitian operators with the orthonormal
eigenfunctions and
and
, then the Fourier
coefficients are given by
|
(1.65) |
In section 1.2.3.4 we have calculated the eigenstates
for an electron in a 3 dimensional trap.
Real electrons are also characterized by the orientation of
their spin which can be either ``up'' () or
``down'' (). The spin-state
of an
electron can therefor be written as
|
(1.66) |
The spins also form a finite Hilbert space
with the orthonormal base
.
If we combine with the solution space for
the spinless problem (equation 1.22), we get a combined
Hilbert space
with the base-vectors
|
(1.67) |
Product States
If we have two independent quantum systems A and B described by
the Hamilton operators and with the orthonormal
eigenvectors and , which are in the
states
|
(1.68) |
then the common state
is given by
|
(1.69) |
Such states are called product states.
Unitary transformations and measurements applied
to only one subsystem don't affect the other as
|
(1.70) |
and the probability to measure the energy
in system A is given by1.11
|
(1.71) |
If
is not a product state, then operations
on one subsystem can affect the other.
Consider two electrons with the common spin state
|
(1.72) |
If we measure the spin of the first electron, we get either
or
with the equal
probability which the resulting post-measurement states
or
.
Consequently, if we measure the spin of the second electron, we
will always find it to be anti-parallel to the first.
Two systems whose common wave-function
is not
a product state are entangled.
Footnotes
- ... abbreviations1.8
-
This formalism is called Braket notation and has been
introduced by Dirac: The
terms are referred to as
``bra''- and the
terms as ``ket''-vectors.
- ... base.1.9
- As physical
observables are real values, their corresponding operators
have to be self-adjoint i.e.
- ... performed.1.10
- since a measurement
can result in a reduction of the wave-function (see
1.3.2.3), it is generally impossible to reconstruct
from the post-measurement state
- ... by1.11
- We assume here
that the eigenvalue isn't degenerated, otherwise
the solution is analog to equation 1.50.
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Up: Quantum Physics in a
Previous: Wave Mechanics
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(c) Bernhard Ömer - oemer@tph.tuwien.ac.at - http://tph.tuwien.ac.at/~oemer/