Phy5646/Problem on Variational Method: Difference between revisions
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Consider the one-dimensional potential | Consider the one-dimensional potential | ||
\psi (x)=\lambda \frac{x^{4}}{4}+\lambda a\frac{x^{3}}{4}-\lambda \frac{a^{2}x^{2}}{8} | \<math>psi (x)=\lambda \frac{x^{4}}{4}+\lambda a\frac{x^{3}}{4}-\lambda \frac{a^{2}x^{2}}{8}</math> | ||
(a) Find the points of classical equilibrium for a particle of mass m moving under the influence of this potential. | |||
(b) Using the variational method, consider the trial wave function | |||
<math>\psi (x)=\left ( \frac{\beta }{\pi } \right )^{\frac{1}{4}}e^\frac{{-\beta (x-x_{0})^{2}}}{2}</math> | |||
<math> | where <math>x_{0}</math> is the global minimum found in (a). Evaluate the expectation value of the energy for this wave function and find the equation defining the optimal values of the parameter β, in order to get an estimate of the ground-state energy. Now take a special, but reasonable, value of the coupling constant, , and obtain the corresponding estimate of the ground-state energy. | ||
Revision as of 21:56, 10 April 2010
Consider the one-dimensional potential
\
(a) Find the points of classical equilibrium for a particle of mass m moving under the influence of this potential.
(b) Using the variational method, consider the trial wave function
where is the global minimum found in (a). Evaluate the expectation value of the energy for this wave function and find the equation defining the optimal values of the parameter β, in order to get an estimate of the ground-state energy. Now take a special, but reasonable, value of the coupling constant, , and obtain the corresponding estimate of the ground-state energy.
