# Finding the Total Energy Equation for Three-Dimensional Free Particle Problems

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 = E* _{x}* + E

*+ E*

_{y}

_{z}With a free particle, the energy of the *x* component of the wave function is

And this equation works the same way for the *y* and *z* components, so here’s the total energy of the particle:

Note that *k*_{x}^{2} + *k*_{y}^{2} + *k*_{z}^{2} is the square of the magnitude of *k* — that is,

Therefore, you can write the equation for the total energy as

Note that because E is a constant, no matter where the particle is pointed, all the eigenfunctions of

are infinitely degenerate as you vary *k** _{x}*,

*k*

*, and*

_{y}*k*

*.*

_{z}