# Quantum Mechanics

### From Chemistry

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## Schrodinger equation

The general case of the Schrodinger equation is written as

There are various ways of expressing this equation and here are some common expressions:

**One Dimensional System:**

- is Planck's constant
- is the mass of the particle
- is the (complex-valued) wavefunction that we want to find
- is a function describing the potential at each point
*x*and - is the energy, a real number.

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## The particle in a 1-dimensional box

- Particle in a box

For the 1-dimensional case in the *x* direction, the time-independent SchrÃ¶dinger equation can be written as:

**The general solutions are:**

**The presense of the walls of the box restricts the acceptable solutions to the wavefunction. At each wall **:

Consider x = 0

- sin 0 = 0, cos 0 = 1. To satisfy D = 0 (cos term is removed)

Now Consider:

- at X=L,
- If C = 0 then for all x and would conflict with Born interpretation
- therefore sin kL must be satisfied by

By substituting k in the energy equation, we get: