Quantum Mechanics/Time Independent Schrödinger

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[edit] Introduction

Consider a particle confined to a one-dimensional box with impenetrable walls. When you solve the Schrödinger equation for the wavefunctions you get two sets of solutions: those of positive parity, and those of negative parity:

and

where n is any positive integer and A is a normalisation constant. Now, we can have all of these infinite states and if you've ever studied Fourier Analysis you may have noticed, with these states you can form any function you wish---that is, the wavefunctions are complete. So what have we learned? Well, a lot actually: we have discovered the eigenstates of the Hamiltonian which can be used to determine the particle's time dependance.

[edit] Derivation of the Time-Independant Schrödinger Equation

We start with the general Schrödinger Equation, and use separation of variables. We have

We separate Ψ into two functions:

Ψ(x,t) = T(t)X(x)

So now the Schrödinger Equation is

We know from earlier that the "interesting" part of the energy operator is a partial derivative with respect to time, and the "interesting" part of the Hamiltonian H is a parrtial derivative with respect to position. As T does not depend on position, it is not affected by H. Similarly, X is not affected by .

So we have:

We can multiply on the left by T ^{− 1}X ^{− 1} to obtain

Note that the left side only depends on x, on the right side only depends on t. We have two functions which are totally indenpendent, but are somehow equal to each other. This is only possible if both functions are equal to a consant, which we call E.

i.e.

X ^{− 1}HX = E

Naturally this implies

HX = EX

and

We can then expand H and and get to solving this thing.