Work Check: Wavefunction Normalisation

[WWCY]

Homework Statement

Find relation between real normalisation constants $B_1$ and $B_2$ for the following wavefunction,
$$
\Psi_k =\sum_{k=1,2} \frac{B_k}{\sqrt{4\sigma ^2 + 2it}} \exp (ip_k (x - \frac{p_k}{2}t) - \frac{(x - p_k t)^2}{4\sigma ^2 + 2it})
$$

The working is rather long so thanks in advance for helping, it is greatly appreciated.

Homework Equations

The Attempt at a Solution

Normalisation requires $\int_{\infty} dx |\Psi (x,0)|^2 = 1$, and
$$\Psi_k (x, 0) =\sum_{k=1,2} \frac{B_k}{2\sigma} \exp (ip_k (x) - \frac{(x)^2}{4\sigma ^2})$$
Knowing that,
$$|\Psi_k|^2 = |B_1 \psi_1|^2 + |B_2 \psi_2|^2 + (B_2 \psi_2)^* (B_1 \psi_1) + (B_2 \psi_2)(B_1 \psi_1)^*$$
$$|B_k \psi_k|^2 = \frac{B_k ^2}{4\sigma^2}\exp (-\frac{x^2}{2\sigma ^2})$$
$$(B_2 \psi_2)(B_1 \psi_1)^* = [\frac{B_1}{2\sigma} \exp (-ip_1 (x) - \frac{(x)^2}{4\sigma ^2})][\frac{B_2}{2\sigma} \exp (ip_2 (x) - \frac{(x)^2}{4\sigma ^2})]$$
$$= \frac{B_1 B_2 }{4\sigma^2} \exp [-i(p_1 - p_2)x - \frac{x^2}{2\sigma^2}]$$
Also,
$$(B_2 \psi_2)^* (B_1 \psi_1) = \frac{B_1 B_2 }{4\sigma^2} \exp [i(p_1 - p_2)x - \frac{x^2}{2\sigma^2}]$$
Before computing the integral, I simplified by letting $\frac{B_1 B_2}{4\sigma ^2} = \phi$, $\frac{1}{2\sigma ^2 } = g$ and $p_1 - p_2 = U$
This gave,
$$(B_2 \psi_2)(B_1 \psi_1)^* = \phi \exp (-gx^2 - iUx)$$
$$(B_2 \psi_2)^*(B_1 \psi_1) = \phi \exp (-gx^2 + iUx)$$
The expression for the integral is,
$$\int_{\infty} dx |\Psi (x,0)|^2 = 1 =\int_{\infty} dx [\frac{B_1 ^2 + B_2 ^2}{4\sigma^2}\exp (-gx^2) + \phi \exp (-gx^2 - iUx) + \phi \exp (-gx^2 + iUx)] $$
The integral of the first integrand can be computed with the Gaussian integral result which gives,
$$\frac{B_1 ^2 + B_2 ^2}{4\sigma ^2} \sqrt{\frac{\pi}{g}} = \frac{B_1 ^2 + B_2 ^2}{4\sigma ^2}\sqrt{2\pi \sigma ^2} = \frac{\sqrt{2\pi} (B_1 ^2 + B_2 ^2)}{4\sigma}$$
The integral of the second integrand, which is also equal to that of the third, was done by completing the square,
$$\int_{\infty} dx \phi \exp (-gx^2 - iUx) = \phi e^{\frac{-U^2}{4g}} \int_{\infty} dx \exp (-[\sqrt{g}x + \frac{iU}{2\sqrt{g}}]^2)$$
I used $dy = \sqrt{g} dx$ and rewrote this as,
$$\frac{\phi e^{\frac{-U^2}{4g}}}{\sqrt{g}} \int_{\infty} dx \exp (-y^2) = \frac{\phi e^{\frac{-U^2}{4g}}}{\sqrt{g}} \sqrt{\pi} $$
Substituting for $g, U, \phi$ then gives,
$$\frac{\sqrt{2\pi} B_1 B_2}{4\sigma} \exp (-\frac{(p_1 - p_2)^2 \sigma ^2}{2})$$

Finally, summing up all 3 integrals of all 3 terms gives,

$$1 =\frac{\sqrt{2\pi} (B_1 ^2 + B_2 ^2)}{4\sigma} + \frac{\sqrt{2\pi} B_1 B_2}{2\sigma} \exp (-\frac{(p_1 - p_2)^2 \sigma ^2}{2})
$$

 

[lekh2003]

Shouldn't this be moved to the Advanced Physics Homework forum?

 

[WWCY]

Shouldn't this be moved to the Advanced Physics Homework forum?

I thought that this was pretty basic stuff (in the context of everything else), and decided to post it here. Apologies if I should have posted this in the other thread.

 

[lekh2003]

I thought that this was pretty basic stuff (in the context of everything else), and decided to post it here. Apologies if I should have posted this in the other thread.

Its not about you not posting in the correct forum. I think you might just get some better and faster answers in the advanced physics forum if what you're studying is wave functions and normalization.

 

[WWCY]

Could anybody take the time to assist? Many thanks!

 

[Ray Vickson]

Homework Statement

Find relation between real normalisation constants $B_1$ and $B_2$ for the following wavefunction,
$$
\Psi_k =\sum_{k=1,2} \frac{B_k}{\sqrt{4\sigma ^2 + 2it}} \exp (ip_k (x - \frac{p_k}{2}t) - \frac{(x - p_k t)^2}{4\sigma ^2 + 2it})
$$

The working is rather long so thanks in advance for helping, it is greatly appreciated.

Homework Equations

The Attempt at a Solution

Normalisation requires $\int_{\infty} dx |\Psi (x,0)|^2 = 1$, and
$$\Psi_k (x, 0) =\sum_{k=1,2} \frac{B_k}{2\sigma} \exp (ip_k (x) - \frac{(x)^2}{4\sigma ^2})$$

***********************************************************************************

$$1 =\frac{\sqrt{2\pi} (B_1 ^2 + B_2 ^2)}{4\sigma} + \frac{\sqrt{2\pi} B_1 B_2}{2\sigma} \exp (-\frac{(p_1 - p_2)^2 \sigma ^2}{2})
$$

Letting $B_1$ and $B_2$ be possibly complex, I used Maple and obtained
$$\int_R |\Psi(x,0)|^2 \, dx = \frac{\sqrt{2 \pi}}{4 \sigma} ( |B_1|^2 + |B_2|^2)
+ \frac{\sqrt{2 \pi}}{4 \sigma} \exp \left( -\frac{1}{2} \sigma^2 (p_1 - p_2)^2 \right) ( B_1 \bar{B_2} + B_2 \bar{B_1})$$
This agrees with your answer if we assume the $B_i$ are real.

 

[WWCY]

Letting $B_1$ and $B_2$ be possibly complex, I used Maple and obtained
$$\int_R |\Psi(x,0)|^2 \, dx = \frac{\sqrt{2 \pi}}{4 \sigma} ( |B_1|^2 + |B_2|^2)
+ \frac{\sqrt{2 \pi}}{4 \sigma} \exp \left( -\frac{1}{2} \sigma^2 (p_1 - p_2)^2 \right) ( B_1 \bar{B_2} + B_2 \bar{B_1})$$
This agrees with your answer if we assume the $B_i$ are real.

Thank you!