- #1

akehn

- 2

- 0

## Homework Statement

Find the variational parameters [itex]\beta[/itex], [itex]\mu[/itex] for a particle in one in one dimension whose group-state wave function is given as:

[itex]\varphi[/itex]([itex]\beta[/itex],[itex]\mu[/itex])=Asin(βx)exp(-[itex]\mu[/itex][itex]x^{2}[/itex]) for x≥0.

The wavefunction is zero for x<0.

## Homework Equations

The Hamiltonian is given as:

H=-[itex]\frac{\\hbar^{2}}{2m}[/itex][itex]\frac{d^{2}}{dx^{2}}[/itex]+V(x)

Where the potential field is defined as follows: V(x)=+∞ for x<0

and V(x)=[itex]\frac{-f}{(x+a)^{2}}[/itex] for x≥0

The terms f, a are positive constants.

3. The Attempt at a Solution

I am familiar with the general procedure. I know that

E(β,μ)=<T> + <V>

Where E are the eigen energies and T, V are the kinetic and potential energies, respectively.

To minimize the Hamiltonian one takes partial derivatives of E with respect to β and μ, setting each term equal to zero to determine the variational parameters. This is all straightfoward.

My confusion lies in the evaluation of the normalization coefficient A from <[itex]\varphi[/itex]|[itex]\varphi[/itex]>=1

The closest tabulated integral has "x" in the exponential term, not "x[itex]^{2}[/itex]"

Any help on solving this integral would be greatly appreciated.