Help Solve for the normalization constant of this QM integral

In summary, the conversation discusses finding the normalization constant for a given wavefunction. However, the integral required to calculate the normalization constant does not converge, indicating that the given wavefunction is not normalizable and therefore not a valid wavefunction. The person also questions whether it could be a typo in the given wavefunction, suggesting a different expression that would make more sense from a quantum mechanical perspective.
  • #1
casparov
30
6
Misplaced Homework Thread
Homework Statement
find A
Relevant Equations
psi = A exp [ - x^2 / (2+ix) ]
I'm given the wavefunction

ψ = A exp(-x^2/(2 + i x))


and I need to find the normalization constant A.

I believe that means to solve the integral

1/A^2 = integral_(-∞)^∞ e^(-x^2/(2 + i x)) e^(-x^2/(2 - i x)) dx


The question does give some standard results for the Gaussian function, also multiplied by x to some different powers in the integrand, but I can't seem to get it into that form.
Whatever I do, I get an x in the denominator of the exponent, and makes it impossible to solve for me.
 
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  • #2
casparov said:
I'm given the wavefunction

View attachment 326989

and I need to find the normalization constant A.

I believe that means to solve the integral

View attachment 326990
That is correct. However, this integral does not converge, so the given wave function is not normalizable (hence not a valid wave function).
 
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  • #3
DrClaude said:
That is correct. However, this integral does not converge, so the given wave function is not normalizable (hence not a valid wave function).
Sorry for the misplacement, it was a question on a final exam. Seems super odd to give this when the follow up questions implied it was normalizable. Thank you very much. Contacted my professor
 
  • #4
casparov said:
Sorry for the misplacement, it was a question on a final exam. Seems super odd to give this when the follow up questions implied it was normalizable. Thank you very much. Contacted my professor
Could it be a typo?
$$
\psi = A \exp \left(- \frac{x^2}{2} + i x \right)
$$
would make more sense from a quantum mechanical point of view.
 
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FAQ: Help Solve for the normalization constant of this QM integral

What is a normalization constant in quantum mechanics?

A normalization constant in quantum mechanics is a factor used to ensure that the total probability of finding a particle within the entire space is equal to one. This is essential because probabilities must sum to one for a physically meaningful wave function.

How do I find the normalization constant for a given wave function?

To find the normalization constant for a given wave function, you need to integrate the absolute square of the wave function over all space and set this integral equal to one. The normalization constant is then determined by solving this equation.

What is the integral form for normalizing a wave function?

The integral form for normalizing a wave function ψ(x) is ∫|ψ(x)|² dx = 1, where the integral is taken over the entire domain of the wave function. This ensures that the total probability is equal to one.

Can you provide an example of solving for a normalization constant?

Sure! For a simple example, consider the wave function ψ(x) = A e^(-αx²). To normalize it, we compute the integral ∫|A e^(-αx²)|² dx over all x. This becomes A² ∫e^(-2αx²) dx. Solving this Gaussian integral, we find A² √(π/(2α)) = 1, so A = (2α/π)^(1/4).

What if the wave function is complex?

If the wave function is complex, you still normalize it by integrating the absolute square of the wave function. For a complex wave function ψ(x) = u(x) + iv(x), where u(x) and v(x) are real functions, you compute ∫|ψ(x)|² dx = ∫(u(x)² + v(x)²) dx and set this equal to one to find the normalization constant.

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