THE MATTER WAVE

The term 'wave' is not well defined. One person's interpretation of a wave will certainly conflict with another. When a surfer says 'he caused the wave to collapse', it is not the same as a quantum wave collapse.

Wikipedia's opening sentences under wave (scientific) is;

In mathematics and science, a wave is a disturbance that travels through space and time, usually accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations

They go on to say;

A single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzy border line.

Whereas the wavefunction is more closely defined as;

A wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle or system of particles. Typically, it is a function of space or momentum or rotation and possibly of time that returns the probability amplitude of a position or momentum for a subatomic particle. Mathematically, it is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.

The wavefunction is a mathematical model of probability that the quantum particle will be found in a particular place.  You could find it there, or it might be elsewhere, but the odds go up in certain locations.

Let's dismantle Schrödinger's equation with a sledgehammer.

i - This is an imaginary number - hold on - it is not like your imaginary friend, this just allows the function to oscillate into negative numbers and still return a possible answer.
h with the cross - This is Planck's constant divided by 2 Pi.
funny a divided by funny a times t - That's really not a funny a, but a rounded d. This represents the partial differential of the energy function with respect to time  The math of calculus allows us to study a relationship of input and output (functions).  So if you wanted to calculate the slope of the Friday night party line using the inputs of a bottle of Jack, beautiful women, and a fast car, then a partial differential at one selected point in time could give you possible outcome solutions.  This is also why you need the imaginary friend and negative numbers.
the pitchfork, 'psi' - This is a statistical odds of finding the particle in a certain place.
the hatted H - the Hamiltonian is the set of possible total energies of the particle.

Two possible solutions for a quantum 'particle in a box' are shown here.  They turn out to be 'standing waves'.  The probability is highest of finding the particle in the location of the greatest deflection, diminishing to zero at node points.

Two possible solutions for a quantum 'particle in a box' are shown here.  They turn out to be 'standing waves'.  The probability is highest of finding the particle in the location of the greatest deflection, diminishing to zero at node points.