Скачать или смотреть Quantum Mechanics - Approximation Methods : Variational Method - One-dimensional Harmonic Oscillator

Using variational method the energy and the corresponding wave functions of a one dimensional harmonic oscillator in its ground state and the first excited state have been estimated. For estimating the ground state energy and eigenfunction the trial function we chose has to be even and smooth everywhere, it must vanish at the boundaries and it must have no nodes. A Gaussian function satisfies these requirements. But what we are not sure about is its width. To account for this, we include in the trial function an adjustable scale parameter. The expectation value of the Hamiltonian is obtained as a function of the adjustable parameter. Minimizing this energy with respect to the adjustable parameter we obtain the energy of the oscillator and the corresponding wave function in its ground state. To calculate the energy and the corresponding wave function for the first excited state we again choose a trial function which must be odd, vanishing at the boundaries, it must have only one node, and it must be orthogonal to the ground state eigenfunction. Repeating the same steps we obtain the energy and the wave function of the oscillator in its first excited state.