Schrödinger equation

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The Schrödinger equation is a mathematical formula that forms the basis of quantum mechanics, the most accurate theory of how subatomic particles behave. It is a mathematical equation that was thought of by Erwin Schrödinger in 1925. It defines something called the wave function of a particle or system (group of particles) which has a certain value at every point in space for every given time. These values have no physical meaning, yet the wave function contains all information that can be known about a particle or system. This information can be found by mathematically manipulating the wave function to return real values relating to physical properties such as position, momentum(mass times velocity), energy, etc. The wave function can be thought of as a picture of how this particle or system acts with time and describes it as fully as possible.

The wave function can be in a number of different states at once, and so a particle may have many different positions, energies, velocities or other physical property at the same time (i.e. "be in two places at once"). However, when one of these properties is measured it has only one specific value (which cannot be definitely predicted), and the wave function is therefore in just one specific state. This is called wave function collapse and seems to be caused by the act of observation or measurement. The exact cause and interpretation of wave function collapse is still widely debated in the scientific community.

For one particle that only moves in one direction in space, the Schrödinger equation looks like:

i\hbar\frac{\partial}{\partial t} \Psi(x,\,t) = \hat H \Psi(x,\,t)

where:

i is the square root of negative one
\hbar is the reduced Planck's constant
t is time
x is a place in space
\Psi(x,\,t) is the wave function
\hat H is the Hamiltonian energy operator:
\hat H \Psi = \hat T \Psi + \hat V \Psi
where:
\hat T is the kinetic energy operator. Which is equal to -\frac{\hbar^2}{2m}(\frac{\partial^2}{\partial x^2} \Psi+ \frac{\partial^2}{\partial y^2} \Psi +\frac{\partial^2}{\partial z^2} \Psi).
\hat V is the potential energy operator equal to V(x), an as yet not chosen function of position.

For one particle that is restricted to a certain region in space (for example: an electron in an atom), the Time-independent Schrödinger Equation can be used, which looks like:

\hat H \Psi = \text{E}\Psi

where:

\text{E} is the energy of the particle

Methods of Separation[change | edit source]

Assuming that the wave function (\Psi (x,t)) is separable, i.e. assuming the function of two variables can be written as the product of two different functions of a single variable:

\Psi (x,t)=\psi (x) T(t)

then, using standard mathematical techniques of Partial Differential equations, it can be shown that the wave equation can be rewritten as two distinct differential equations given by:

i\hbar \frac{1}{T(t)} \frac{d^2 T(t)}{dt^2}=C

and

-\frac{\hbar^2}{2m} \frac{d^2 \psi (x)}{dx^2}+V(x) \psi (x)=C

where the first equation is solely dependent on time T(t), and the second equation depends only on position \psi (x), and where C is just a number. Generalizations can easily be made for higher numbers of spacial dimensions (up till 3 dimensions of space and 1 of time).

Interpretations of the Wave function[change | edit source]

Born Interpretation[change | edit source]

There are many philosophical interpretations of the wave function, and a few of the leading ideas will be considered here. The main idea, called the Born probability interpretation (named after physicist Max Born) comes from the simple idea that the wave function is square integrable; i.e.

\int_{-\infty}^\infty \! |\Psi (x,t)|^2 dx < \infty

This rather simple formula has great physical implications. Born hypothesized that The above integral determines that the particle exists somewhere in space. But how can we find it? We use the integral

\int_b^a \! \Psi (x,t) dx=P_{b<x<a}

where P_{b<x<a} is the probability of finding the particle in the region from b to a. In other words, the particle is only a probability, and you can never know the location of the particle. Basically, this is the Born interpretation.

Copenhagen Interpretation[change | edit source]

An extension of the above ideas can be made. Since the Born interpretation says that the actual position particle cannot be known, we can derive the following. If \Psi_1, \Psi_2, \Psi_3,...\Psi_n are solutions to the wave equation, then the superposition of those solutions, i.e.

\Psi_s=c_1\Psi_1 + c_2\Psi_2 + c_3\Psi_3 + ... c_n\Psi_n

is also a solution. This implies, then, that the particle exists in every possible position. When an observer comes and measures the position of the particle, then the superposition is reduced to a single possible wave function. (i.e., \Psi_s\Psi_n, where \Psi_n is any of the possible wave function states.) This idea that a particle's position cannot exactly be known, and that a particle exists in multiple positions simultaneously gives rise to the Uncertainty principle. The mathematical formulation of this principle can be given by

\Delta x\Delta p>\frac{\hbar}{2}

Where \Delta x is the uncertainty in position, and \Delta p is the uncertainty in momentum. This principle can be mathematically derived from the Fourier transforms between momentum and position as defined by quantum mechanics, but we will not derive it in this article.

Other Interpretations[change | edit source]

There are various other interpretations, such as the many-worlds interpretation, and quantum determinism.