In quantum physics, you can use the Schrödinger equation to see how the wave function for a particle in an infinite square well evolves with time. The Schrödinger equation looks like this:
![image0.png](https://www.dummies.com/wp-content/uploads/397821.image0.png)
You can also write the Schrödinger equation this way, where H is the Hermitian Hamiltonian operator:
![image1.png](https://www.dummies.com/wp-content/uploads/397822.image1.png)
That’s actually the time-independent Schrödinger equation. The time-dependent Schrödinger equation looks like this:
![image2.png](https://www.dummies.com/wp-content/uploads/397823.image2.png)
Combining the preceding three equations gives you the following, which is another form of the time-dependent Schrödinger equation:
![image3.png](https://www.dummies.com/wp-content/uploads/397824.image3.png)
And because you’re dealing with only one dimension, x, this equation becomes
![image4.png](https://www.dummies.com/wp-content/uploads/397825.image4.png)
This is simpler than it looks, however, because the potential doesn’t change with time. In fact, because E is constant, you can rewrite the equation as
![image5.png](https://www.dummies.com/wp-content/uploads/397826.image5.png)
That equation makes life a lot simpler — it’s easy to solve the time-dependent Schrödinger equation if you’re dealing with a constant potential. In this case, the solution is
![image6.png](https://www.dummies.com/wp-content/uploads/397827.image6.png)
Neat. When the potential doesn’t vary with time, the solution to the time-dependent Schrödinger equation simply becomes
![image7.png](https://www.dummies.com/wp-content/uploads/397828.image7.png)
the spatial part, multiplied by
![image8.png](https://www.dummies.com/wp-content/uploads/397829.image8.png)
the time-dependent part.
So when you add in the time-dependent part to the time-independent wave function, you get the time-dependent wave function, which looks like this:
![image9.png](https://www.dummies.com/wp-content/uploads/397830.image9.png)
The energy of the nth quantum state is
![image10.png](https://www.dummies.com/wp-content/uploads/397831.image10.png)
Therefore, the result is
![image11.png](https://www.dummies.com/wp-content/uploads/397832.image11.png)
where exp (x) = ex.