· Nearly any system near equilibrium can be approximated as a H.O.
· One of a handful of problems that can be solved exactly in quantum mechanics
Solve diff. eq.: General solutions are sin and cos functions
or can also write as
where A and B or C and f are determined by the initial conditions.
spring is stretched to position x0 and released at time t = 0.
Spring oscillates with frequency
and maximum displacement from equilibrium
Energy of H.O.
Kinetic energy º K
Potential energy º U
Total energy = K + U = E
Most real systems near equilibrium can be approximated as H.O.
e.g. Diatomic molecular bond
For small deviations from eq.
Total energy of molecule in 1D
COM coordinate describes translational motion of the molecule
QM description would be free particle or PIB with mass M
We’ll concentrate on relative motion (describes vibration)
and solve this problem quantum mechanically.
Note: replace m with m (reduced mass) if
Goal: Find eigenvalues En and eigenfunctions yn(x)
This is not a constant, as it was for P-I-B,
so sin and cos functions won’t work.
TRY: (gaussian function)
which matches our original diff. eq. if
We have found one eigenvalue and eigenfunction
This turns out to be the lowest energy: the “ground” state
For the wavefunction, we need to normalize:
where N is the normalization constant
Note is symmetric. It is an even function:
There are no nodes, & the most likely value for the oscillator displacement is 0.
So far we have just one eigenvalue and eigenstate. What about the others?
These have the general form
Note E increases linearly with n.
Þ Energy levels are evenly spaced
regardless of n
There is a “zero-point” energy
E = 0 is not allowed by the Heisenberg Uncertainty Principle.
Symmetry properties of y’s
are even functions
are even functions
Useful properties: (even) ×(even) = even
(odd) ×(odd) = even
(odd) ×(even) = odd
Чтобы распечатать файл, скачайте его (в формате Word).
Ссылка на скачивание - внизу страницы.