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I'm a second-semester junior who is studying chemistry and mathematics. I've self-studied basic quantum chemistry before, and I picked up Griffith's Introduction to Quantum Mechanics a few days ago to get a broader/more rigorous taste of the subject. I've been going through the reading/problems, and I'm really confused about something...

For those who want to follow along in the book, I'm referring to question 2.8 in the first edition.

Problem: Given [tex]\Psi(x,0)=Ax(a-x)[/tex] for a particle in a 1-dimensional box [tex](0<x<a)[/tex], find the time-dependent wave function for any given time.

Solution:

[tex]\Psi(x,t)=\sqrt\frac{2}{a}\sum^{\infty}_{n=1}c_n e^{\frac{-in^2 \pi^2 \hbar t}{2ma^2}}} sin\frac{n\pi x}{a}[/tex]

I found that:

[tex]c_n = \frac{\sqrt{60}}{a^3}\int^{a}_{0}x(a-x)sin\frac{n\pi x}{a}dx = \frac{8\sqrt{15}}{n^3\pi^3} [/tex]

... if n is odd. Its 0 if n is even.

Thus, the wavefunction is equal to:

[tex]\Psi(x,t)=\frac{4\sqrt{30}}{\pi^3n^3\sqrt{a}}\sum^{\infty}_{n=1}(1-(-1)^n)e^{\frac{-in^2 \pi^2 \hbar t}{2ma^2}}} sin\frac{n\pi x}{a}[/tex]

I know that I could rewrite that sum over just the odd integers. But it doesn't matter that much for now.

My questions are:

1. How can a particle in a box actually have such an initial wavefunction? I was under the impression that a particle in a box MUST have a wavefunction described by the simple sinusodidal waves that you get by solving the time-independent Schrodinger equation!

2. If a particle in a box can have any initial wavefunction, doesn't that mean that it can also have any initial energy? Where does the concept of quantization of energy come into play in this analysis?

3. Does the particle have a definite energy? How would I verify that its constant, as it should be?

Sorry for the confusion. Quantum chemistry books hardly ever cover time-dependence in quantum mechanics! I'm going to analyze my solution further. However, it's 4:40 AM and I should go to bed :)

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# Analysis of time-dependent SE (infinite square potential)

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