According to Quantum Mechanics,
all particles in the Universe are described by what we call a wave function.
Properties such as position and momentum
do not have defined values until they are observed.
The probability of each possible observation is determined by the wave function.
Suppose we have a particle moving freely through space in one dimension.
Its wave function may look as shown.
The probability of the particle being at a particular position is given by
the square of the amplitude of the wave function at that location.
Now let us consider a particle that is moving in one dimension,
but is trapped inside a box.
Its wave function may look as shown.
At each point in space and time, all wave functions are described by a complex number.
All complex numbers have a real component and an imaginary component.
A particle with a higher energy will have a wave function
that rotates with a higher frequency, and it will look like this.
The wave function can also be in a combination of this waveform and the previous one.
We say that this wave function is in a superposition
of the two wave functions we saw earlier.
Since this waveform is a combination of two different wave functions
with different energy levels, the energy of this particle is uncertain.
When we measure the energy,
we say that the wave function collapses into one of the two original wave functions.
The wave function’s frequency determines the particle’s energy.
The momentum of the particle is determined by the wave function’s wavelength.
A longer wavelength implies a smaller momentum.
In this scenario, we have a particle moving freely through space with a known momentum.
However, the position of the particle is completely unknown,
because the amplitude of the wave function is the same everywhere.
The square of the amplitude of the wave function at each location
determines the probability of measuring the particle at that location.
We can have some knowledge of the particle’s location
if the wave-function consists of the sum of several different waveforms
with different wavelengths.
In this case, we have more knowledge of the particle’s location,
since the amplitude of the wave function is larger in one area.
But, we now have less knowledge of the particle’s momentum,
since the wave function is composed of waveforms of different wavelengths.
If we measure the momentum precisely,
then the wave function will collapse into one of these waveforms,
and we will again have no knowledge of the particle’s position.
The more knowledge we have about the particle’s momentum,
the less knowledge we have about the particle’s position.
And the more knowledge we have about the particle’s position,
the less knowledge we have about the particle’s momentum.
This is because
the more we want to limit the wave function’s peak in amplitude to one narrow location,
the greater the number of different wavelengths we will have to add together.
Note that the velocity of the particle is not the velocity of any of the
individual waveforms, but the velocity of the pattern that forms
out of the sum of all of the waveforms that make up the total wave function.
When a particle is traveling freely through space,
there are no limitations to a wave-function’s possible wavelength and frequency.
On the other hand, if a particle is trapped inside a box,
the only wavelengths and frequencies that are possible are the ones that ensure
that the amplitude of the wave function is close to zero at the boundary of the box.
This means that for a trapped particle, waveforms in between these frequencies
are not allowed, and hence only certain energy levels are possible.
At each point in space and time, the wave function is described by a complex number
which has a real component and an imaginary component.
In order to help visualize the upcoming explanations,
let us just show the real component of the wave-function.
So far, we have only been talking about particles moving in one dimension.
Now, let us consider a particle moving in two dimensions.
Amplitude must always be close to zero along all four edges.
If the particle is trapped inside a box,
then only certain energy levels will be possible in each of the two dimensions.
The frequency in each of the two dimension indicates the energy in that dimension.
Energy must be present in both dimensions to ensure zero amplitude at the edges.
Now let us consider a particle called an electron, moving in three dimensions,
trapped by the electrical attraction of an atomic nucleus.
As in the previous examples,
only certain energy levels for the electron will be possible.
The wave functions with specific energy levels that are possible
are what we refer to as the electron orbitals of an atom.
Much more information on Quantum Mechanics
is available in the other videos on this channel.
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