If we have a quantum particle moving freely through space, its wave function may look as shown.
On the other hand, if we have a quantum particle trapped inside a box, its wave function may look as shown.
Why do the wave functions look so different in the two cases?
To answer this question, let us consider a wavefunction where the amplitude is the same everywhere throughout all of space.
Because the amplitude is the same everywhere, we have no knowledge of the particle’s position.
But, we know the momentum with 100% certainty, and this momentum is determined by the wavelength.
A particle with a larger momentum is described by a wavefunction with a shorter wavelength.
The wave function we saw at the beginning of this video is actually a superposition of several different wavefunctions with different wavelengths.
The larger the number of wavefunctions of different wavelengths we add together, the larger the peak in amplitude will be in one particular location, which gives us more knowledge about the probability of the particle’s position.
But, this also causes us to have less knowledge of the particle’s momentum, because the particle now has a probability of being measured with the momentum associated with each of the various wavelengths.
Let’s now consider the case of a particle, with a known energy, trapped inside a box.
Here, there is uncertainty if the momentum is moving to the right, or if the momentum is moving to the left.
A particle with a known momentum moving to the right has a wavefunction that looks like this blue spiral.
A particle with a known momentum moving to the left has a wavefunction that looks like this orange spiral.
Since we don’t know if the momentum is to the left or to the right, the total wavefunction for the particle is the sum of the blue spiral and the orange spiral.
The sum of the two spirals is shown here in green.
This gives us the wavefunction for the trapped particle, which looks like a flat, rotating, sine wave.
This satisfies the boundary condition that the amplitude of the wavefunction must be close to zero at the boundary of the box.
Whereas a quantum particle travelling freely through space can have any wavelength, trapped quantum particles can only have wavelengths where the amplitude is close to zero at the boundary of the box.
In this video, we have only discussed the case of a single particle.
Things get much more exciting when we have multiple identical particles trapped together, and this is discussed in the video, "What causes the Pauli Exclusion Principle?"
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