Quantum mechanics, free particle normalization question

[Jdraper]

Homework Statement

A free particle has the initial wave function

ψ(x,0)=Ae^(-a|x|)

Where A and a are positive real constants.

a) Normalize ψ(x,0)

Homework Equations

1= ∫|ψ|^2 dx

The Attempt at a Solution

I attempted to normalize using 1= ∫|ψ|^2 dx from -∞ to ∞. When doing this i obtained

1=(A^2)∫e^(-2a|x|) dx from -∞ to ∞. doing this integral between these limits i get 0 as the value of the integral, which is obviously wrong.

I looked up the answer online as this is a problem from Griffiths, introduction to quantum mechanics. (problem 2.21 in the link)

http://www.thebestfriend.org/wp-content/uploads/IntroductiontoQuantumMechanics2thEdition.pdf

The solution on there says you integrate from 0 to ∞, this is then my question, why do you integrate from 0 to ∞ instead of from -∞ to ∞?

Thanks, any help would be appreciated.

John

 

[rwooduk]

you have to split the integral because for minus infinity, plus infinity limits it is divergent (i.e. does not exist)

here's a useful tool to check this

http://www.integral-calculator.com/#

it will also give you 1/2a as the answer for integrating exp(-2ax) between 0 and infinity. which is what you need.

edit also see this page on improper integrals...

http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx

 

[Jdraper]

I think I get it, is it like the integral of x^3 from -1 to 1 = 0? I'm guessing that's why the factor of 2 appears in the model answer. Anyway thank you, I believe I get it now.

 

[Oxvillian]

Wait!

It's not a divergent integral - notice the absolute value sign in the exponent.

Since it's an even function that's being integrated, you can integrate from $0$ to $\infty$ and then double the answer.

 

[vela]

Homework Statement

A free particle has the initial wave function

ψ(x,0)=Ae^(-a|x|)

Where A and a are positive real constants.

a) Normalize ψ(x,0)

Homework Equations

1= ∫|ψ|^2 dx

The Attempt at a Solution

I attempted to normalize using 1= ∫|ψ|^2 dx from -∞ to ∞. When doing this i obtained

1=(A^2)∫e^(-2a|x|) dx from -∞ to ∞. doing this integral between these limits i get 0 as the value of the integral, which is obviously wrong.

I looked up the answer online as this is a problem from Griffiths, introduction to quantum mechanics. (problem 2.21 in the link)

http://www.thebestfriend.org/wp-content/uploads/IntroductiontoQuantumMechanics2thEdition.pdf

The solution on there says you integrate from 0 to ∞, this is then my question, why do you integrate from 0 to ∞ instead of from -∞ to ∞?

Thanks, any help would be appreciated.

John

You didn't evaluate the integral correctly if you got 0. You shouldn't get different answers which depend on the method you choose to integrate. If you're getting inconsistent results, it means you're making a mistake.