Next: 7.5 Quantum-Classical Correspondence
Up: 7. The Harmonic oscillator
Previous: 7.3 Taylor series solution
The polynomial solutions occur for
= (n + ). |
(7.21) |
The terminating solutions are the ones that contains only even
coefficients for even n and odd coefficients for odd n.
Let me construct a few, using the relation (7.16).
For n even I start with a0 = 1, a1 = 0, and
for n odd I start with a0 = 0, a1 = 1,
| H0(y) |
= |
1, |
(7.22) |
| H1(y) |
= |
y, |
(7.23) |
| H2(y) |
= |
1 - 2y2, |
(7.24) |
| H3(y) |
= |
y - y3. |
(7.25) |
Question: Can you reproduce these results? What happens if I start with
a0 = 0, a1 = 1 for, e.g., H0?
In summary:
The solutions of the Schrödinger equation occur for
energies
(n +
)
, an
the wavefunctions are
(In analogy with matrix diagonalisation one often speaks of eigenvalues
or eigenenergies for E, and eigenfunctions for
.)
Once again it is relatively straightforward to show how to normalise
these solutions. This can be done explicitly for the first few polynomials,
and we can also show that
 (x) (x) dx = 0 if n1
n2. |
(7.27) |
This defines the orthogonality of the wave functions. From a more
formal theory of the polynomials Hn(y) it can be shown that
the normalised form of
(x) is
(x) = 2-n/2(n!)-1/2
exp
- x2 Hn x |
(7.28) |
Next: 7.5 Quantum-Classical Correspondence
Up: 7. The Harmonic oscillator
Previous: 7.3 Taylor series solution