Imaginary Normalisation Constant

[Ruddiger27]

Homework Statement

A one-dimensional system is in a state at time t=0 represented by:

Q(x) = C { (1.6^0.5)Q1(x) - (2.4^0.5)Q2(x)}

Where Qn(x) are normalised eergy eigenfunctions corresponding to different energy eigenvalues, En(n=1,2)

Obtain the normalisation constant C

The Attempt at a Solution

I get C= i(1.2)^0.5 from the following equation:

C^2 * (1.6 (int( Q1 ^2 dx) - 2.4(int ( Q2 ^2 dx = 1

So C^2 has to be -5/4 in order for the above to be true. Is this right?
Just a bit confused over whether it's possible to have an imaginary value for the normalisation constant? Thanks for any help you can give.

 

[Allday]

You haven't formed the product Q(x)Q*(x) correctly. What is special about energy eigenstates?

 

[Ruddiger27]

they follow linear superposition? so the integral of the total wavefunction squared is equal to the integral of 1.6*Q1^2 plus the integral of 2.4*Q2^2?

 

[dextercioby]

What is Q* equal to ? How do you define the scalar product?

Daniel.

 

[Ruddiger27]

Well there aren't any imaginary parts to the first wavefunction since its just in the form Q = C ( XQ1 - YQ2) so Q* is just the same as Q.

 

[Gokul43201]

they follow linear superposition?

What do you know about the integral of Q1Q2*?

so the integral of the total wavefunction squared is equal to the integral of 1.6*Q1^2 plus the integral of 2.4*Q2^2?

Yes. Are you absolutely clear why this is so?

 

[Ruddiger27]

I would think the integral of Q1Q2* would be zero since these wavefunctions are orthogonal, so I would end up with C^2 =5/20.