Wavefunction question

I have a question that says;

A particle living in one spatial dimension has a wavefunction (function f(x) in this case) given by;

f(x) = Ae^{-|x|/a}

It asks me to find the normalisation constant, so I ASSUMED you would take limits over infinity and minus infinity and integrate

|f(x)|^{2}

, with that equal to one. If I do this I get

[\frac {-A^{2}ae^{-2|x|/a}}{2}]

= 1 with limits infinity and minus infinity.

As you can tell, that answer just gives me zero on the left hand side. What exactly am I meant to do?

Reply 1

Scipio90

You need to consider x > 0 and x <0 separately.

Reply 2

Anti Elephant Mine

Original post by Scipio90

You need to consider x > 0 and x <0 separately.

Right, do you mean integrate with limits infinity to 0, and 0 to minus infinity then add them together? I've checked that and it still gives me zero <_>;

Reply 3

Scipio90

Yes. Post your working.

Reply 4

Anti Elephant Mine

Original post by Scipio90

Yes. Post your working.

[\frac {-A^{2}ae^{-2|x|/a}}{2}]

limits infinity and 0 is;

[\frac {-A^{2}ae^{-2|\infty|/a}}{2} - \frac {-A^{2}ae^{-2|0|/a}}{2}] = [0 - \frac {-A^{2}a}{2}] = [\frac {A^{2}a}{2}]

[\frac {-A^{2}ae^{-2|x|/a}}{2}]

limits 0 to minus infinity is;

[\frac {-A^{2}ae^{-2|0|/a}}{2} - \frac {-A^{2}ae^{-2|-\infty|/a}}{2}] = [\frac {-A^{2}a}{2} - \frac {-A^{2}ae^{-2|\infty|/a}}{2}] = [\frac {-A^{2}a}{2} - 0] = [\frac {-A^{2}a}{2}]

Which added together just becomes zero.

Reply 5

Scipio90

Sketch e^-|x|. Is your answer for the <0 part sensible?

Reply 6

Anti Elephant Mine

Original post by Scipio90

Sketch e^-|x|. Is your answer for the <0 part sensible?

Aye, the previous question asked me to sketch the wave equation. It's reflected in the y-axis meaning the area should be double of what is on either side. But why am I getting zero :|. I have a feeling that the integration of e^-2|x|/a is different to what I've put.