Quantum Mechanics: An Introduction
The deBroglie relation is central to the conceptual understanding of
nature of quantum mechanics. Everything, particles and waves obey the
relation:
It immediately becomes clear that the distinction between matter and light is
not clear, as matter and radiation possess both particulate (corpuscular) and
wavelike (undulatory) properties. Quantum mechanics also immediately
requires us to carefully inquire about the properties of our system.
Classically, we can easily ask for the position of a particle. But what is the
position of a wave? That is only 'defined' for some finite fraction of the
wavelength itself. Suppose we guess the uncertainty (knowing the answer
makes this easier):
Then
If we try to minimize the momentum and still know the position we can reach
a minimum 'uncertainty' limit
Uncertainty in position and momentum is quantified by the Root Mean
Square (RMS) deviation from the mean value:
We learn about the position and momentum of system by examining the
expectation values of observables, which correspond to the average values
(<x>, for example) in the equation above. We will talk more about how the
expectation value is obtained, below.
The momentum and position are conjugates of each other. Conjugate
variables cannot be simultaneously determined for a system in a given
state. The act of measuring one of the conjugate variables changes the
value one obtains by measuring the other. In fact, the measurement of both
position and momentum, or any other pair of non-commuting variables)
changes the state of the system, no matter how carefully you perform the
measurements. The mere statement that 'one
precisely knows the values of the position and momentum variables' in malformed
and not suited to quantum mechanics.
In QM, all observables correspond to operators
Another set of conjugates are energy and time. They have an uncertainty
relation too, then
The operators that correspond to conjugate have a finite (non-zero)
commutator
Schroedinger described a system quantum mechanically by defining
the coordinate representation form for the operators for position and
momentum. Here:
But what do the operators operate on? Schroedinger went
on to describe a material system with a
wave function which entered into
a wave equation (in analogy to a classical wave mechanics). What
he really did was apply an example of an eigenvector - eigenvalue equation:
The operator that he chose to emphasize is the one which corresponds to
the total energy, the Hamiltonian. The total energy of an isolated system is
conserved.
The total energy is the sum of the Potential, V, and Kinetic, T,
energies. The Hamiltonian, H, is then
If we use the momentum operator defined above and assume a constant
(zero) potential, then
which is the time-independent Shroedinger equation for a free particle.
E is a scalar constant (number)
which is equal to the numerical value of the total energy of the system
described by the wavefunction 
.
Just as in classical mechanics, the energy is relative to an arbitrary choice of zero.
In the coordinate representation, we
can solve for as in terms of x
because the SE is a second order
differential equation in x. The solution to such an equation has two particular
[sic] solutions and any solution is a linear combination of those two.
The solution to the free particle SE is then:
where
Solution exists for any value of E. Quantum mechanics does not quantize
the energy of a free particle, only one that is constrained to be confined in a
finite region of space (bound systems).
A Note on notation: I am writing the wavefunctions (eigenvectors of
Schroedingers equation) in a slightly different way than you will find in your
text. For the moment we may make the following coordinate space
representation identifications:
The probability of finding the particle anywhere is then:
where the integration is over all space.
The set of eigenvectors of a Hermetian Operator (eq. the Hamiltonian) are
orthonormal and complete
Top