Page 47 - Science at the nanoscale

RPS: PSP0007 - Science-at-Nanoscale
9:2
June 9, 2009
3.2. Basic Postulates of Quantum Mechanics
associated with the physical observable as follows:
Z
∗
ψ QψdV
(3.10)
hqi =
Note that the wavefunction is assumed to be properly normalised
If the wavefunction is
and the integration is over all space.
represented as a linear combination of the eigenfunctions of the
operator Q, then the above operation would give rise to the possi-
ble values for the physical observables multiplied by a probability
coefficient. Hence this is essentially a weighted average of the pos-
sible observable values.
For a physical system that is free of external interactions, the
evolution of the physical system with time is given by
h dψ
(3.11)
Hψ = i
2π dt
√
−1 and H is the Hamiltonian operator. This equa-
where i =
tion is derived from the classical Hamiltonian with the substitu-
tion of the classical observables by their corresponding quantum
mechanical operators. The role of the Hamiltonian is contained in
the Schr¨odinger equation.
3.2.1
Schr¨ odinger Equation
As we have seen thus far, the wavefunction for a physical sys-
tem contains everything there is to know about the system. How 37 ch03
do we find the exact form of this wavefunction? There are many
different varieties of dynamical systems, so how do we find the
wavefunction that corresponds to the dynamical problem? Erwin
Schr¨odinger formulated an equation that allows the wavefunc-
tion to be determined for any given physical situation. The
Schr¨odinger equation is the analogue of Newton’s equation in
Classical Mechanics.
The time dependent Schr¨odinger equation is given as follows
h 2 2 h dψ
− ∇ ψ + Vψ = i (3.12)
8mπ 2 2π dt