# John Lindsay Orr

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# Notes on the Schrödinger Equation

The two key ingredients are the Planck equation for the energy of a photon, and the de Brogli9e wavelength of a matter wave.

The Planck equation for the energy, $E$ of a photon with frequency $f$ is:

The de Broglie wavelength, $\lambda$, of the matter wave asoociated with a particle with momentum $p$ is:

It's convenient to recast this in terms of the wave number $k = 2\pi / \lambda$ and the normalized frequency $\omega = 2\pi f$. (Conceptually, $k$ represents the number of wave cycles in an interval of length $2\pi$, and $\omega$ represents the numbers of cycles in a time interval of length $2\pi$.) In these terms, the Planck equations and the de Broglie wavelength formulas become respectively where $\hbar = h/2\pi$ is the adjusted Planck constant, approximately $1.05 \times 10^{-34} J\cdot s$.

The Schrödinger Equation is motivated by looking for a wave-type equation which has as a solution the simplest possible candidate for a matter wave, i.e., a plane wave of the form

Then compute and so

Next we calculate and then, using the classical (non-relativistic) relationship together with the de Broglie formula, we obtain

Putting together the two formulas for $E\psi$ we obtain the free Schrödinger Equation:

Finally, since we are interpreting as the energy operator for a free particle, it makes sense that the energy operator for a particle in potential $V(x, t)$ is and so we reassemble the equations for a particle in a potential as