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I'm following Griffiths' Introduction to Quantum Mechanics, and I see that he's got 2 different expressions for $\Psi(x,t)$. One of them is $$\Psi(x,t)=\psi(x)\exp\left(\frac{-iEt}{\hbar}\right)\tag{2.7}$$. The other is $$\Psi(x,t)=\sum\limits_{n=1}^{\infty} c_n \psi_n(x)\exp\left(\frac{-iE_nt}{\hbar}\right)\tag{2.18}$$

Whenever I am solving a problem, I tend to try to use Eqn 2.18, but then I look up the solution and see that I'm supposed to use Eqn 2.7 instead. It seems as if 2.18 would be the more complete equation as it is referred to as the most general solution. However, Griffiths keep using 2.7 in most problems that I've come across. I find it strange that he's used the same notation on the left hand side to refer to two different equations. Which one is the wavefunction?

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1 Answer 1

2.18 is simply a linear superposition of wavefunctions represented in 2.7. 2.7 represents one particular eigenfunction and 2.18 is the general solution with linear superposition over all possible eigenfunctions. They should both solve the Schrodinger equation. They are both correct wavefunctions, and 2.18 is the more general case (to get 2.7, just set the coefficients to 0 for all the irrelevant terms).

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