# 2. Variational Principle. The energy of a system with wave function ψ is given by where...

###### Question:

2. Variational Principle. The energy of a system with wave function ψ is given by where H is the energy operator. The variational principle is a method by which we guess a trial form for the wave function φ, with adjustable parameters, and minimize the resulting energy with respect to the adjustable parameters. This essentially chooses a "best fit" wave function based on our guess. Since the energy of the system with the correct wave function will always be minimum our guess will always lead to an energy which is slightly too high, but the variational principle allows us to get as close as possible to the correct energy with our trial wave function Use the variational principle to estimate the ground state energy for the anharmonic oscillator, 2m 2 Compare your result with the exact result 1m Note that this is a one-dimensional problem, so take dV-dx. You know physically that a trial function must be peaked around x-0 and must be normalizable (i.e.. Jo o ψ2ax is finite). Such functions would include e-e or 1/(c +x2), or the harmonic oscillator ground state wave function for instance. Choose wisely, and the mathematics will be far simpler.

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