The Quantum Wave Function Explained
In Quantum mechanics particles are things we see only when they are measured. There movement patterns are described by a wave function that satisfies the Schrodinger Equation. Wave functions are no unique to Quantum mechanics and are used in other systems such as motion for the ripples of water, sound waves, vibrations on a string, electro magnetic waves etc. Each of these systems has there own wave equation that have similarities as they all express the change of the wave function in space and time. The key difference though is that the quantum wave function isn’t a physical wave. There is no medium so there is no way to prove that they are real with our current knowledge but they describe the behaviors of quantum particles very well.
So lets do the next best thing and visualize it! Here is the equation and it depends on the factors of time and space.
This equation uses 1 dimension of space which is a simplification as a real particle is in 3 dimensions of space but this simplification allows us to plot the function. So lets do that. Lets start off by plotting the first term:
What we get is simply a cos wave. Note that k is the variable controlling the size of the wave length, the omega controls the oscillation frequency and A controls the amplitude.
To plot the second part of the wave we are going to need another axis even though the wave is one dimensional. This is because the sin part of the wave function is being multiplied by the imaginary number i which makes it a complex wave function.
The following image shows the two waves plottd idependently:
Now lets see what happens when we plot the whole thing together:
We end up with a spiral of complex values from negative infinity to positive infinity. The following image is fixed at one point in time but if time continued to move the resulting wave would oscillate around the x — axis, similar to a bar inside of a turning coil.
Now that we have understood what a quantum wave function looks like, lets tackle the more fun question. What is it?
Well, its called a probability amplitude, and isn’t anything physical on its own but if you take the mod square of the amplitude it tells you the probability of finding the particle at any point in time in this one dimensional space. It also tells you about all other measurable physical properties by doing different mathematical operation on each one.
Overall the wave function is a mathematical tool that keeps track of all the properties of a quantum particle and explains our observations of the probabilistic nature of where the particles appear.
Currently we have only explored one example of a wave function but there are so many more examples. They just have to satisfy a set of constraints.
- A wave function must be a solution to the Schrodinger Equation
- The wave function must be normalizable: When you calculate a probability from a wave function you will get a probability distribution. The area of this distribution must equal 1 because there has to be a definite probability that you’ll find the particle somewhere. This means that the originals wave function can’t have an infinite area. This means that our visualization from earlier is not technically allowed because it goes from negative infinity to positive infinity. To be valid it would have to go to 0 as x goes to infinity.
- The wave function must be single valued.
- It must be continuous
- The slope of the wave function must be continuous.
Finally lets look at super position which is not a property unique to quantum. If you have two sets of ripples in water that overlap, any point will feel the two waves added together. This is the super position of waves. In quantum if you have two or more wave functions that are valid solutions of the schrodinger equation then any combination of these wave functions is also a valid solution. This is where the idea comes that the schrodinger’s cat can be dead and alive at the same time.
Although this never really happens due to other quantum phenomena’s such as entanglement and decoherence. These concepts along with the EPR paradox by Einstein will be explored in future articles.
