Time evolution of the state vector
The time evolution of the state vector is prescribed by the Schrdinger equation
where H is the Hamiltonian operator. This equation can be solved, in principle,
yielding
where
is the initial state vector. The operator
is
Quantum system physical state
The physical state of a quantum system is represented by a vector denoted
which is a column vector, whose components are probability amplitudes for different
states in which the system might be found if a measurement were mad
Physical Observables
Physical observables are represented by linear, hermitian operators that act on the
vectors of the Hilbert space. If A is such an operator, and
the Hilbert space, then A might act on
as
is an arbitrary vector in
to produce a vector
,
The Heisenberg uncertainty principle
Because the operators X and P are not compatible,
, there is no
measurement that can precisely determine both X and P simultaneously. Hence, there
must be an uncertainty relation between them that specifies how uncerta
The Heisenberg picture
In all of the above, notice that we have formulated the postulates of quantum
mechanics such that the state vector
evolves in time but the operators
corresponding to observables are taken to be stationary. This formulation of quantu
depends on all components of
. If we write the expectation values as integrals (in
one-dimension, for example), then we see that
which shows that
depends on all values of the function
, which is known as
a trial wave function. We, therefore, call
a functi
Eigenvectors
By multiplying both sides of this equation by
and using the orthonormality
condition, it can be seen that the expansion coefficients are
The eigenvectors also satisfy a closure relation:
where I is the identity operator.
Averaging over many i
Variational Theory and the Variational
Principle
A very useful approximation method is known as the variational method. This is the
basis of much of quantum chemistry, including Hartree-Fock theory, density
functional theory, as well as variational quantu