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SCHRODINGER WAVE EQUATION DERIVATION:

Schrödinger's equation follows very naturally earlier developments:

In 1905, by considering the photoelectric effect, Albert Einstein had published his

$E = h f\;$

formula for the relation between the energy E and frequency f of the quanta of radiation (photons), where h is Planck's constant.

In 1924 Louis de Broglie presented his de Broglie hypothesis which states that all particles (not just photons) have an associated wavefunction $\Psi\;$ with properties:

$p=h / \lambda\;$ , where $\lambda\,$ is the wavelength of the wave and p the momentum of the particle.

De Broglie showed that this was consistent with Einstein's formula and special relativity so that

$E = h f\;$

still holds, but now this is hypothesized to hold for all particles, not just photons anymore.

Expressed in terms of angular frequency $\omega = 2\pi f\;$ and wavenumber $k = 2\pi / \lambda\;$ , with $\hbar = h / 2 \pi\;$ we get:

$E=\hbar \omega$

and

$\mathbf{p}=\hbar \mathbf{k}\;$

where we have expressed p and k as vectors.

Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:

$\psi \approx e^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}$

and to realize that since

$\frac{\partial}{\partial t} \psi = -i\omega \psi$

then

$E \psi = \hbar \omega \psi = i\hbar\frac{\partial}{\partial t} \psi$

and similarly since:

$\frac{\partial}{\partial x} \psi = i k_x \psi$

then

$p_x \psi = \hbar k_x \psi = -i\hbar\frac{\partial}{\partial x} \psi$

and hence:

$p_x^2 \psi = -\hbar^2\frac{\partial^2}{\partial x^2} \psi$

so that, again for a plane wave, he got:

$p^2 \psi = (p_x^2 + p_y^2 + p_z^2) \psi = -\hbar^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \psi = -\hbar^2\nabla^2 \psi$

And by inserting these expressions into the Newtonian formula for a particle with total energy E, mass m, moving in a potential V:

$E=\frac{p^2}{2m}+V$ (simply the sum of the kinetic energy and potential energy; the plane wave model assumed V = 0)

he got his famed equation for a single particle in the 3-dimensional case in the presence of a potential:

$i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$

Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, $\psi\;$ , moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a series of four papers in 1926.

The Schrödinger equation defines the behaviour of $\psi\;$ , but does not interpret what $\psi\;$ is. Schrödinger tried unsuccessfully to interpret it as a charge density. In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted $\psi\;$ as a probability amplitude, although Schrödinger was never reconciled to this statistical or probabilistic approach.

In the mathematical formulation of quantum mechanics, a physical system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a ray in that space. The nonzero elements of a Hilbert space are by definition normalizable and it is convenient, although not necessary, to represent a state by an element of the ray which is normalized to unity. This vector is often somewhat loosely referred to as wave function, although in a more rigorous formulation of quantum mechanics a wave function is a special case of a state vector. (In fact, a wave function is a state in the position representation, see below). A state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. It contains all information of the system that is knowable in a quantum mechanical sense. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.

In Dirac's bra-ket notation at time

t

the state is given by the ket $|\psi(t)\rangle$ . The time-dependent Schrödinger equation, giving the time evolution of the ket, is:

$H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{d}{d t} \left| \psi \left(t\right) \right\rangle$

where

i

is the imaginary unit,

t

is time,

d / d t

is the derivative with respect to

t

, $\hbar$ is the reduced Planck's constant (Planck's constant divided by $2\pi\,$ ), $\psi(t)\,$ is the time dependent state vector, and

H (t)

is the Hamiltonian (a self-adjoint operator acting on the state space). If one assumes a certain representation for $\psi\,$ , for instance position or momentum representation, the state vector is assumed to depend on more variables than time alone, and the time derivative must be replaced by the partial derivative $\partial / \partial t.$

The Hamiltonian describes the total energy of the system. As with the force occurring in Newton's second law, its form is not provided by the Schrödinger equation, but must be independently determined from the physical properties of the system.

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