Understanding the Wave Equation and Separation of Variables for Particle States

[kasse]

A particle is in a state described by $$(\frac{mk}{\pi^2 \hbar^{2}})^{1/8}exp(- \frac{1}{2 \hbar} \sqrt{mk}x^{2})exp(-if(t))$$

When applying separation of variables here, my book ignores the first fraction and sets

$$g(x) = exp(- \frac{1}{2 \hbar} \sqrt{mk}x^{2})$$

$$h(t) = exp(-if(t))$$

But then $$\Psi(x,t) \neq g(x)h(t)$$ right?

 

[Hootenanny]

That is indeed correct.

 

[kasse]

The book also says that $$p_{op} = -i \sqrt{mk}x exp(\frac{-1}{2 \hbar}\sqrt{mk}x^2)$$, so it has ignored the first fraction again. Is this wrong?

 

[Hootenanny]

What is P_op meant to represent?

 

[kasse]

The momentum. $$p_{op}=-i \hbar \frac{\partial}{\partial x}$$

 

[Hootenanny]

Which text is this from? Looking at the wave function, the momentum should be a function of time and should include the nomalisation constant as you say.

 

[kasse]

An exam problem at my university, so it oughtn't be wrong.