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**1. Homework Statement**

Determine the constant λ in the wave equation

[itex]\Psi(x) = C(2a^2 x^2 + \lambda)e^{-(a^2 x^2/2)}[/itex]

where [itex]a=\sqrt{mω/\hbar}[/itex]

**2. Homework Equations**

Some standard integrals I guess

**3. The Attempt at a Solution**

So I believe the wave equation just needs to be normalised. Using the usual conditions for normalisation,

[itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} | x^2 e^{-(a^2 x^2/2)} + e^{-(a^2 x^2/2)} |^2 dx =1[/itex]

From there,

[itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} |2x^2 e^{-(a^2 x^2/2)}|^2 dx =1[/itex]

Then squaring the function inside the integral and moving the '4' outside the integral as it is a constant,

[itex]4(C2a^2 + C\lambda)^2 \int^{∞}_{-∞} x^4 e^{-(a^2 x^2)} dx =1[/itex]

Now that should be a standard integral but I don't know any involving an x term to the fourth power. Or perhaps I've done something else wrong?