- #1
akehn
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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.