At some point, your quantum physics instructor may want you to find the total energy equation for three-dimensional free particle problems. The total energy of the free particle is the sum of the energy in three dimensions:
E = Ex + Ey + Ez
With a free particle, the energy of the x component of the wave function is
![image0.png](https://www.dummies.com/wp-content/uploads/397224.image0.png)
And this equation works the same way for the y and z components, so here’s the total energy of the particle:
![image1.png](https://www.dummies.com/wp-content/uploads/397225.image1.png)
Note that kx2 + ky2 + kz2 is the square of the magnitude of k — that is,
![image2.png](https://www.dummies.com/wp-content/uploads/397226.image2.png)
Therefore, you can write the equation for the total energy as
![image3.png](https://www.dummies.com/wp-content/uploads/397227.image3.png)
Note that because E is a constant, no matter where the particle is pointed, all the eigenfunctions of
![image4.png](https://www.dummies.com/wp-content/uploads/397228.image4.png)
are infinitely degenerate as you vary kx, ky, and kz.