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 abbreviations^{1.8}

(1.24) |

(1.25) |

The scalar product of the eigenfunctions and from the on dimensional capacitor example (1.2.3.3) gives

(1.27) |

(1.28) |

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) |

(1.30) |

(1.31) |

This describes a standard sine-

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) |

(1.45) |

(1.46) |

(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) |

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.

**Measured Values:**Measured values are always eigenvalues of their according operator .**Probability Spectrum:**If the eigenvalue isn't degenerated and has the eigenvector , then the probability to measure is . If the eigenvalue is -fold degenerated and is an orthonormal base of the according eigenspace, then

**Reduction of the Wave Function:**If the eigenvalue isn't degenerated, the post-measurement state , otherwise

(1.51)

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) |

(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) |

(1.55) |

(1.56) |

(1.57) |

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) |

(1.59) |

(1.60) |

(1.61) |

Unitary Operators

The operator of temporal evolution satisfies the condition

(1.62) |

Unitary operators can also be used to describe abstract operations
like rotations

(1.63) |

(1.64) |

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) |

(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) |

(1.69) |

(1.70) |

(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) |

Two systems whose common wave-function
is not
a product state are *entangled*.

- ... abbreviations
^{1.8} -
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
- ... by
^{1.11} - We assume here that the eigenvalue isn't degenerated, otherwise the solution is analog to equation 1.50.