Anharmonic Oscillator
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 63
14-8-2016
1230
Anharmonic Oscillator
Use variational methods in one dimension to estimate the ground state energy of a particle of mass m in a potential V(x) = Ax4.
SOLUTION
Many possible trial functions can be chosen for the variational calculation. Choices such as exp (-α|x|) are poor since they have an undesirable cusp at the origin. Instead, the best choice is a Gaussian:
(1)
where the potential in the problem is
(2)
We evaluate the three integrals in (A.3.1)–(A.3.4).
(3)
(4)
(5)
(6)
We have used (A.3.1) to derive the last expression. Now we find the minimum energy for this choice of trial function by taking the derivative with respect to the variational parameter α. Denote by α0 the value at this minimum:
(7)
(8)
(9)
This result for E0 is higher than the exact eigenvalue.
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