- #1

SamRoss

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- Homework Statement
- A particle in the infinite square well has its initial wave function an even mixture of the first two stationary states:

Ψ(x,0) = A[##\psi_1 (x) + \psi_2 (x)##]

If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them?

- Relevant Equations
- Ψ(x,0) = A[##\psi_1 (x) + \psi_2 (x)##]

##\sum_{n=1}^\infty |c_n|^2=1##

I am working through David Griffiths' "Introduction to Quantum Mechanics". All of the solutions are provided online by Griffiths himself. This is Problem 2.5(e). I understand his solution but I'm confused about one thing. After normalizing Ψ, we find ##A=\frac {1}{\sqrt2}##. Griffiths notes that the probability of obtaining each of the energies associated with each ##\psi## is 1/2 which makes sense. What I'm confused about is what the coefficients ##c_n## in front of the ##\psi_n## are (every wave function Ψ is supposed to be a linear combination of the ##\psi_n##). If we distribute ##A=\frac {1}{\sqrt2}## in Ψ(x,0) = A[##\psi_1 (x) + \psi_2 (x)##] then it seems like the coefficients ##c_n## are each just ##\frac {1}{\sqrt2}## which would be nice because then ##\sum_{n=1}^\infty |c_n|^2## (all ##c_n## being equal to 0 for n>2) would equal 1 which it should. But aren't the ##c_n## supposed to be

*different*from A? This leads me to look at Ψ(x,0) = A[##\psi_1 (x) + \psi_2 (x)##] again and conclude that each coefficient is simply 1, but this wouldn't work because then ##\sum_{n=1}^\infty |c_n|^2## would be equal to 2. So what are the ##c_n##?