AltSci Cell

by Javantea
April 8-9, 2025

This problem comes from Griffiths’ Introduction to Quantum Mechanics. It’s problem 2.1a.

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*Problem 2.1 Prove the following theorems:

(a) For normalizable solutions, the separation constant E must be real. Hint: Write E (in Equation 2.6) as E₀ + iΓ (with E₀ and Γ real), and show that if Equation 1.20 is to hold for all t, Γ must be zero.

The wave function

Ψ(x,t) = ψ(x)e^−iEt/ℏ

Assuming E is complex.

E = E₀ + iΓ

We normalize our wave function.

∫_−∞^+∞|Ψ(x,t)|²dx = 1

|Ψ(x,t)|² = ψ(x)²e^−2iEt/ℏ

Since we are integrating by x and not t, the whole right side comes out of the integral.

e^−2iEt/ℏ∫_−∞^+∞ψ(x)²dx = 1

I’m pretty sure this is already wrong because I’m not doing the |Ψ|. That is.. yeah instead of multiplying it by itself I should be multiplying by the complex conjugate. So let’s get some computer algebra to help us along with this.

Using Gemini: see normalize_gemini1.md

"When you square the absolute value of a complex number, |z|² = z * conjugate(z), where conjugate(z) is the complex conjugate of z (obtained by changing the sign of the imaginary part)."

So I was right about that.

Gemini and maxima agree that abs(Psi(x,t))² = psi(x)² but I am now confused.

So now we integrate.

∫psi(x)² = 1

Nowhere here do we see E. So we’re stuck. So we ask gemini.

Do you agree with the statement "For normalizable solutions, the separation constant E must be real."? Why?

Okay Gemini is repeatedly contradicting itself, so let’s go through it’s final logic.

$$\begin{gathered} E = E_r + iE_i \\ e^{-iEt/\hbar} = e^{-i(E_r + iE_i)t/\hbar} \\ = e^{-iE_rt/\hbar}e^{-i^2E_it/\hbar} \\ = e^{-iE_rt/\hbar}e^{E_it/\hbar} \\ |e^{-iEt/\hbar}|^2 = e^{-iEt/\hbar} * conjugate(e^{-iEt/\hbar}) \\ conjugate(e^{-iEt/\hbar}) = e^{iE_rt/\hbar}e^{E_it/\hbar} \\ |e^{-iEt/\hbar}|^2 = e^{-iE_rt/\hbar}e^{E_i)t/\hbar}e^{iE_rt/\hbar}e^{E_it/\hbar} \end{gathered}$$

So now we can simplify our answer here.

$$\begin{gathered} |e^{-iEt/\hbar}|^2 = e^{-iE_rt/\hbar}e^{E_it/\hbar}e^{iE_rt/\hbar}e^{E_it/\hbar} \\ = e^{-iE_rt/\hbar + E_it/\hbar + iE_rt/\hbar + E_it/\hbar} \\ = e^{E_it/\hbar + E_it/\hbar} \\ = e^{2E_it/\hbar} \end{gathered}$$

Now it’s possible to confidently solve the problem. With a bit of logic that Gemini used (that we understood but obviously it’s not completely honest to say that this thought is original when Gemini said it first), we can explain the nature of the normalization.

∫_−∞^+∞|Ψ(x,t)|²dx = 1

 = e^2E_it/ℏ∫_−∞^+∞ψ(x)²dx = 1

No matter what our ψ is, setting t = ∞, we get

 = e^2E_i∞/ℏ∫_−∞^+∞ψ(x)²dx = 1

The only legitimate answer to this where E_i is not zero is where ψ(x) cancels out the infinity it is being multiplied to. I am not aware of any non-zero solution and since we know that the whole thing has to be equal to 1, we have a pretty clear solution to this problem.

E_i = 0

Conclusion

This problem took me a very long time to solve mainly because I went to Gemini and Maxima with it. The only reason I went to Gemini was because I was having some problem with complex numbers, so it makes sense when a problem involves complex numbers (such is true of every single problem in Quantum Mechanics), I am unfortunately going to need to improve my understanding and not simply ask Gemini for help.

Gemini showed a complete and total understanding of the problem that is very impressive -- immediately contradicting itself when the problem is asked in a different way. This is because Quantum Mechanics assumes that E is real. This problem actually proves that E is real. This is a major problem in its reasoning and the way that I found to get it to help me solve this problem was by pointing out its contradiction. Pointing out its contradiction required me to mostly understand the problem. This seems to indicate that the solution to this problem was somewhere inside me the whole time. Indeed using Gemini's vast intelligence to work on this problem showed me where that was and resulted in a profound discovery that will certainly compound.

Those who are bearish on Artificial Intelligence, LLMs, and so on look at these sort of mistakes and say "see I told you so". But if you look at what happened in this discussion, you can see that Gemini is making mistakes because it was never trained to be a physicist. It's doing Quantum Mechanics at the level of a person. That is not to say that it is going to come up with correct answers.