See the Math Probability Density

The wavefunctions that describe stationary states of quantum mechanical systems have the mathematical form:

math symbols

where math symbols(x) is a real function, and Et/math symbols is called the complex phase of the wavefunction. Each variable has a physical meaning: x is position, E is the system's energy, and t is time. Using Euler's formula (see sidebar), we can rewrite this as:

math symbols

Thus, the quantum wavefunction has both a real part and an imaginary part. The probability density that we would observe in an experiment, however, is proportional to the absolute value of the wavefunction squared:

math symbols

Which can be rewritten like this:

math symbols

math symbols

math symbols

In the first line, we have used the definition of absolute value squared, in the second line we have multiplied the sines and cosines together, and in the third line we have applied the fact that sin2(x) + cos2(x) = 1.

The experimentally observed probability density, therefore, has no imaginary part. It is real and positive, as expected. However, the imaginary part of the wavefunction can make a huge difference in how two quantum mechanical objects interact, as we have seen in the quantum interference of particles in Unit 5 and will see in the quantum interference of BECs in this unit.