# Variational Method

A particle is moving in one dimension under a potential $$V(x)$$ such that, for large positive values of $$x$$, $$V(x) \approx kx ^\beta$$, where $$k>0$$ and $$\beta$$ $$\geq$$ 1. If the wave function in this region has the form $$\psi(x)\sim \exp(-x^\lambda)$$, Then show that, $$\lambda=\frac{\beta}{2}+1.$$

I am not sure how to approach this problem. I thought of using Variational Method, but ended up in a complicated Gaussian integral. Is there any other approach as to reach to this result.

• What have you tried? Provide your working on the problem and it will greatly help people try and spot specifically where you went wrong. Commented May 12, 2022 at 19:15
• Have you tried plugging this wave function in the time-independent Schrödinger equation ? Commented May 12, 2022 at 20:20

We can use one dimensional time-independent Schrödinger equation, $$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\Psi(x)+V(x)\Psi(x) = E\Psi(x)$$
Given that $$\Psi(x)\approx e^{-{x^\lambda}}$$ and $$V(x)\approx kx^\beta$$, so we get
$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}e^{-{x^\lambda}}+kx^\beta e^{-{x^\lambda}} = Ee^{-{x^\lambda}}$$ which gives $$-\frac{\hbar^2}{2m}\lambda[(\lambda-1)x^{\lambda-2}-\lambda x^{2(\lambda-1)}]+kx^{\beta}= E$$ Now we assume $$\lambda>0$$, then $$O(x^{\lambda-2}), so we may neglect the first term in the limit of large x. We can now write, $$\frac{\hbar^2}{2m}\lambda^2 x^{2(\lambda-1)}+kx^{\beta}\approx E$$ Differentiating both sides wrt x we get $$2\frac{\hbar^2}{2m}\lambda^2 (\lambda-1) x^{2(\lambda-1)-1}+k\beta x^{\beta-1}\approx 0$$ Now notice that the RHS of the equation is 0 while LHS is x dependent. So this equation can only be valid for all x if $$2(\lambda-1)-1=\beta - 1$$ because if the two are not equal then both the x terms will be linearly independent and their linear combination cannot yield zero. Finally we get, $$\lambda = \frac{\beta}{2}+1$$