2010 PDE2 Schrödinger Equation

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Kyle's presentation from 2010_01_29 started to explain motivations for the time dependent Schrödinger equation of quantum mechanics. Please find below a brief note I wrote for myself about this topic.Colliand 22:03, 30 January 2010 (UTC)

Momentum of a pure wave

Consider a (pure, monochromatic) wave $$\psi (t,x) = e^{-i(\omega t - k \cdot x)}$$ with temporal frequency $\omega$ and spatial wave 3-vector $k$. The parameters $k $ and $\omega$ encode the oscillation properties of this pure wave. In his theory of black body radiation, Planck associated the notion of energy to a wave with his relation $$ E = \hbar \omega. $$ Later, Einstein, in his theory of the photoelectric effect, demonstrated the usefulness of this energy-frequency relationship.

The Planck relation $E = \hbar \omega$ is inconsistent with special relativity. This inconsistency led de Broglie to a more general relationship spawning a link with momentum and the wave vector $k$. It turns out that the complete phase $(\omega t - k \cdot x)$ is Lorentz invariant but the time-frequency $\omega$ is not. Since energy is a physical quantity which should not depend upon reference frame, we encounter the troubling inconsistency.

The resolution of the inconsistency involves enlarging our viewpoint to account for the relativistic linkage between space and time. The wave 4-vector $(\frac{\omega}{c}, k) = {\bf{K}}$ is Lorentz invariant. L. de Broglie observed that the momentum 4-vector ${\bf{P}} = ( \frac{E}{c}, p)$ is also Lorentz invariant and argued we should improve the Planck relationship to ${\bf{K}} = c_0 \bf{P}$. Thus, wave properties encoded in the wave 4-vector $\bf{K}$ are associated with classical mechanics particle notions encoded in $\bf{P}$. The Planck relation identifies the constant of proportionality: $c_0 = \hbar$.

We thus obtain the de Broglie formula $$ p = \hbar k $$ linking momentum to wave vector.

Momentum of a general wave? Schrödinger's Motivation

Special relativitistic inconsistency of Planck's wave-energy relationship $E = \hbar \omega$ led de Broglie to the consistent 4-vector relationship $\bf{K} = \hbar {\bf{P}}$. This led to the (also relativistically inconsistent) de Broglie wave-momentum relationship $p = \hbar k$. Thus, we have a dictionary for pure monochromatic waves which links wave properties $(\omega, k)$ to dynamical constructs from classical physics $(E, p)$. Schrödinger's ${\bf{idea}}$ was to relax to considering more general waves (e.g. a superposition of plane waves) without a well-defined wave vector $k$ or frequency $\omega$ and to then extract the physical notions of momentum and energy from the wave by some other means.

• What do you have to do to the plane wave $e^{-i(\omega t - k \cdot x)}$ to "get" the energy $E = \hbar \omega$. Well, we apply the operator $i \hbar \partial_t \psi = \hbar \omega \psi$. (Think of the $\psi$ as coming along for the ride and the operation $i \hbar \partial_t$ as something you do to the wave to read off the energy.)
• What do you have to do to the plane wave $e^{-i(\omega t - k \cdot x)}$ to "get" the momentum $p = \hbar k$? Similarly, you apply the operator $\frac{\hbar}{i} \nabla$ to $\psi$.

Schrödinger's Equation

The energy-momentum to frequency-wave vector relationship for pure waves motivated Schrödinger to introduce operators which read off the energy and momentum of general waves. Schrödinger knew that classical mechanics encodes deterministic evolution equations using relationships among energy and momentum. For example, a particle of mass $m$ in a potential well given by $V(x)$ has the energy: $$ E = \frac{|p|^2}{2m} + V(x). $$ Using the dictionary $E \rightarrow i \hbar \partial_t, ~ p \rightarrow \frac{\hbar}{i} \nabla$ acting on the wave function $\psi$, we translate the energy-momentum equation of the classical particle into $$ i \hbar \partial_t \psi = - \frac{\hbar^2}{2m} \Delta \psi + V(x) \psi, $$ Schrödinger's equation of quantum mechanics!