2.1 Principles of quantum mechanics

The theory of quantum mechanics is built upon the fundamental concepts of
* wave-functions* and * operators*. The wave-function is a
single-valued square-integrable function of the
system parameters and time which provides a complete description of the system.
Linear Hermitian operators act on the wave-function and
correspond
to the physical * observables*, those dynamical variables which can be
measured, e.g. position, momentum and energy.

For systems of atomic nuclei^{2.1} and electrons, which are
the subject of this dissertation, the system parameters might be taken to be
a set of position variables of the constituent particles (the notation
adopted in this and the following chapters is to refer to electronic variables using a latin index and
nuclear variables with a greek index) i.e.
, their momenta
or even a mixture of the two e.g.
. In contrast to a Newtonian system which is
completely described by the positions * and* momenta of its
constituents, the quantum-mechanical wave-function is a function of only
one of these parameters per particle^{2.2}. The wave-function for the system is thus
typically
denoted by
.

A notation due to Dirac [4] is often employed, which reflects the
fact
that this wave-function is simply one of many representations of a single
* state-vector* in a Hilbert space, which is written as
,
known as
a * ket*. There also exists a * dual space* containing a set of
* bra* vectors, denoted
, defined by their scalar
products and in one-to-one correspondence with the kets.
The scalar product is written as a * braket* and is anti-linear in the
first argument and linear in the second: thus
.
It is worth noting here that state-vectors which differ only by
a multiplicative non-zero complex constant describe the same state: we can thus
restrict our interest to the set of * normalised* vectors defined such
that the scalar product of the vector with its own conjugate equals unity:

(2.1) |

The operator corresponding to some observable is often written ,
and in general when this operator acts on some state-vector
,
a different (not necessarily normalised) state-vector
results:

(2.2) |

in which the constant (always real for Hermitian operators) is the

The postulates of quantum mechanics [5] state that for a system in state :

- the outcome of a measurement of a dynamical variable is always one of the eigenvalues of the corresponding operator,
- immediately following a measurement, the state-vector collapses
to the eigenstate
corresponding to the measured eigenvalue
^{2.3}, - the probability of such a measurement
^{2.4}is

Much of the power of the theory comes from the fact that the quantum-mechanical
states can be linearly superposed since this leads to no ambiguity in the action
of linear operators^{2.5}.
We now consider the quantity
. From Sturm-Liouville theory, the eigenstates of the
operator form a * complete set*, which means that any valid
state-vector can be expressed as a linear superposition of those
eigenstates with appropriate complex coefficients :

Either taking scalar products of both sides of equation 2.5 with the eigenstates , or by using the following concise expression of completeness;

(2.7) |

Now this result is applied to the quantity
:

(2.10) |

in which we have used the fact that is linear, that the are eigenstates of (2.3) and the orthonormality relation (2.6).

Since the only possible outcomes of a measurement of the observable
corresponding to operator are the eigenvalues
,
with corresponding probabilities
(2.4), the quantity
is to be interpreted as the
* expectation value* of for a system in state
.
The normalisation condition
corresponds to
the condition that the probabilities sum to unity.

The final postulate of quantum mechanics states that between measurements, the state-vector
evolves in time according to the time-dependent Schrödinger
equation^{2.6}:

(2.11) |

in which the nuclear masses and atomic numbers appear. The first two terms on the right-hand side represent the kinetic energies of the electrons and nuclei respectively. The subsequent terms describe the electron-nuclear, electron-electron and inter-nuclear Coulomb interaction energies respectively.

Finally we note that if we solve the time-independent Schrödinger
equation, the eigenvalue equation for the Hamiltonian, then the
time-dependence of the wave-function takes a particularly simple form. The
following separation of variables is made:

(2.13) |

The ordinary differential equation 2.15 is straightforwardly solved, so that eigenfunctions of the Hamiltonian with energy take the form:

(2.16) |

(2.17) |

From now on we shall be dealing with eigenstates of the Hamiltonian, and so will suppress the exponential time-dependence of the state and deal directly with the time-independent state instead.