We now introduce the central equation of quantum mechanics, the Schrödinger equation, which can be seen as a generalisation of the phenomenological explanations described above to situations where particles experience arbitrary forces. We also introduce the concept of operators related to observables (measurable properties such as momentum and energy).
For simplicity consider a particle moving in one dimension, with position
measured by a coordinate
(7) |
For light (photons),
(8) |
Indeed, if we insert
(9) |
Let us multiply the left-hand side of Eq. (9) with
(10) |
For arbitrary
(11) |
This reformulation turns out to be very convenient. Notice that the operations
(12) | |||
(13) |
allow us to directly read off the energy and momentum of the photons with
wavefunction (7). If
(14) | |||
(15) |
the result always fulfils the wave equation
(16) |
Notice the similarity to Eq. (10). We call
Points to remember
Physical observables are associated with operators.
Momentum operator
Energy operator
Position operator
E.g.:
For light, we have
For particles with finite mass
(at non-relativistic speed), the total energy is given by
Reading the correspondence principle given above backwards, one finds the Schrödinger equation
(17) |
Let us decode this equation and express it in usual mathematical notation:
(18) |
In the steps above
we associate to the potential energy an operator
(19) |
It is only consequent to also introduce the position operator
(20) |
and the operator of kinetic energy
(21) |
is called the Hamilton operator (or Hamiltonian), which is of central importance in quantum mechanics. It corresponds to the total energy in classical mechanics.
We now can restate the Schrödinger equation in the compact form
(22) |
Even more compactly we can write
(23) |
where we
suppressed the arguments.
Points to remember
The evolution of the wave function (=state of the system) for a
quantum particle of mass
(24) |
or
(25) |
where
If
(26) |
where the time dependence
(27) |
Divide by the time-dependent factor
(28) |
where
(29) |
This is the most important equation in this module, and we will
study it for a wide range of problems. Hence, remember it well!
Points to remember
For a particle of mass
(30) |