According to Quantum Mechanics,
all particles in the universe are described by what we call a wave function,
and quantum wave functions are described by Schrodinger’s Equation.
Although the behavior of all subatomic particles is inherently probabilistic,
Schrodinger’s equation does not itself contain any probabilities.
The probability of every possible observation is determined by the wave function,
but prior to the observation,
the wave function changes and evolves in a completely deterministic manner.
This deterministic way in which a wave function behaves
without any observer present is what Schrodinger’s Equation describes.
Charged particles exert forces on one another.
We can view this by saying that at different points in space,
a charged particle has a different amount of potential energy.
The height of this graph at each point in space is represented by the variable “V”.
The height does not represent a spatial dimension,
but the value of the potential energy for our charged particle at each location.
“M” represents the mass of the particle.
“H bar” represents Planck’s constant divided by two pi.
This variable represents the value of the wave function at each point in space and time.
This graph shows a wave function along just one of the three spatial dimensions.
At each point in space and time, the wave function is a complex number
with a real a component and an imaginary component.
This expression indicates
how much the wave function changes as X increases by a small amount.
This change is a complex number which can be represented with arrows as shown.
Without changing the X coordinate or direction of any of these arrows,
let us show each of these arrows starting along the X axis.
The amount by which these arrows change as we increase X by a small amount
is represented with this other expression.
At each point in space, the sum of these three values
is a complex number represented with the following symbol.
We have now covered all the symbols on the left part of Schrodinger’s equation.
Let us now focus on the right side of the equation.
As in the previous example, this graph represents the value of a wave function
along just one of the three spatial dimensions.
These arrows indicate how much the wave function at each point in space will change
if the time increases by a small amount.
Without changing the X coordinate or direction of any of these arrows,
let us show each of these arrows starting along the X axis.
If we multiply these arrows by “h bar”, their length changes.
If we then multiply these arrows by the imaginary number “i”,
all these arrows are rotated by 90 degrees.
Each of these arrows now represent a complex number
which is equal to the right side of Schrodinger’s equation.
The right side of Schrodinger’s equation must always equal the left side.
If a particle is moving in just one dimension,
then Schrodinger’s equation can be simplified to the following.
If we have a function with a shorter wavelength, then this parameter is larger.
For both sides of the equation to be equal, this other parameter must also be larger.
This other parameter represents how quickly the wave function is rotating.
Therefore, if we have a wave function with a shorter wavelength,
the wave function must rotate with a higher frequency.
The higher the frequency, the higher the energy of the particle.
If the energy of the particle is known, and is represented by the variable “E”,
then the right side of Schrodinger’s equation is equal to the following.
Therefore, if the energy of a particle is known with certainty,
Schrodinger’s equation can be simplified as shown.
If we also know the potential energy at each point in space,
symbolized by the variable “V”,
we can then calculate all the possible wave functions for the particle.
Much more information on this topic is available
in the video “Quantum Wave function Visualization.”
Also, details about how a wave function determines the probabilities
for all possible observations is available in the video “Quantum Operators.”
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