Schrödinger's Equation

A classical particle of mass $m$, moving in a one-dimensional potential $U(x)$, satisfies the energy conservation equation

$$E = K+ U,$$ (4.13)

where

$$K = \frac{p^{2}}{2\,m}$$ (4.14)

is the particle's kinetic energy. (See Sections 1.3.2 and 1.3.5.) Hence,

$$E\,\psi = (K+U)\,\psi$$ (4.15)

is a valid, but not obviously useful, wave equation.

However, it follows from Equations (4.10) and (4.11) that

$$\frac{\partial \psi}{\partial t} = -{\rm i}\,\omega\,\psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)} = -{\rm i}\,\frac{E}{\hbar}\,\psi,$$ (4.16)

which can be rearranged to give

$$E\,\psi= {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}.$$ (4.17)

Likewise, from Equations (4.10) and (4.12),

$$\frac{\partial\psi}{\partial x} = {\rm i}\,k\,\psi_0 \,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)} = {\rm i} \,\frac{p}{\hbar}\,\psi,$$ (4.18)

which can be rearranged to give

$$p\,\psi= -{\rm i}\,\hbar\,\frac{\partial\psi}{\partial x}.$$ (4.19)

It immediately follows that

$$p^2\,\psi = -\hbar^2\,\frac{\partial^2\psi}{\partial x^{\,2}}.$$ (4.20)

Hence,

$$K\,\psi=\frac{p^{2}}{2\,m}\,\psi = -\frac{\hbar^{2}}{2\,m}\,\frac{\partial^{2}\psi}{\partial x^{2}}.$$ (4.21)

Thus, combining Equations (4.15), (4.17), and (4.21), we obtain

$${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^{2}}{2\,m}\,\frac{\partial^{2}\psi}{\partial x^{2}} + U(x)\,\psi.$$ (4.22)

This equation, which is known as Schrödinger's equation—because it was first formulated by Erwin Schrödinder in 1926—is the fundamental equation of wave mechanics.

For a massive particle moving in free space (i.e., $U=0$), the complex wavefunction (4.10) is a solution of Schrödinger's equation, (4.22), provided

$$\omega = \frac{\hbar}{2\,m}\,k^{2}.$$ (4.23)

The previous expression can be thought of as the dispersion relation for matter waves in free space. (See Section 4.2.6.) The associated phase velocity (i.e., propagation speed of a wave maximum) is

$$v_p = \frac{\omega}{k} = \frac{\hbar\,k}{2\,m} = \frac{p}{2\,m},$$ (4.24)

where use has been made of Equation (4.12). However, this phase velocity is only half the classical velocity, $v=p/m$, of a massive (non-relativistic) particle.