The circumstance and the life route forced me to stay away from quantum mechanics for more than twenty years. Now, I do not know my ABC in quantum mechanics. A friend with just a high school level of math knowledge asked me to explain the Schrodinger equation and the wave function in simple words. I have nothing for myself so I cannot deliver anything to others.
My friend is a mobile library and a sophisticated professional in other field. So the question is not posed by a casual and ignorant folk.
How To Answer The Question?
Thank You,
ABDUL-SAHIB
Typically, by “Schrödinger equation”, we mean the differential equation giving the time-evolution of the wave function that, at some initial time, describes the state of the examined quantum system. In other words, the Schrödinger equation is the equation of motion in quantum mechanics; it is the analogue of the Newton’s second law in classical mechanics.
However, there also exists “another” Schrödinger equation, the stationary or time-independent, Schrödinger equation, which is abusively also referred to as the Schrödinger equation.
The stationary Schrödinger equation is the most important equation in quantum mechanics, because if we solve it, we have solved the examined quantum system.
The stationary Schrödinger equation is the differential equation giving the energies of the quantum system and the respective wave functions that describe states of constant energy – or energy eigenstates – of the system.
Having found the energies and the respective wave functions of the system, then any possible state of the system is described by a linear combination of the wave functions describing the energy eigenstates of the system.
On the other hand, the wave function is a real-valued complex function that has no physical meaning; it is a mathematical tool, but the square of the absolute value of the wave function gives the probability density for the examined system, i.e. it gives the probability to find the examined system in a small neighborhood around a particular point in space.
The wave function is a means to an end: computing the probability density. The Schrödinger equation is just the evolution equation for the wave function. What matters is the equation that describes the evolution of the probability density with time. This is a continuity equation, where the current density has a simple expression in terms of the wave function, but a complicated expression in terms of the density. That's why using the wave function is useful and solving the Schrödinger equation is easier than solving the continuity equation directly. But what matters is the probability density.
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