# 7.3 The schrӧdinger equation

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By the end of this section, you will be able to:
• Describe the role Schrӧdinger’s equation plays in quantum mechanics
• Explain the difference between time-dependent and -independent Schrӧdinger’s equations
• Interpret the solutions of Schrӧdinger’s equation

In the preceding two sections, we described how to use a quantum mechanical wave function and discussed Heisenberg’s uncertainty principle. In this section, we present a complete and formal theory of quantum mechanics that can be used to make predictions. In developing this theory, it is helpful to review the wave theory of light. For a light wave, the electric field E ( x , t ) obeys the relation

$\frac{{\partial }^{2}E}{\partial {x}^{2}}=\frac{1}{{c}^{2}}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}E}{\partial {t}^{2}},$

where c is the speed of light and the symbol $\partial$ represents a partial derivative . (Recall from Oscillations that a partial derivative is closely related to an ordinary derivative, but involves functions of more than one variable. When taking the partial derivative of a function by a certain variable, all other variables are held constant.) A light wave consists of a very large number of photons, so the quantity ${|E\left(x,t\right)|}^{2}$ can interpreted as a probability density of finding a single photon at a particular point in space (for example, on a viewing screen).

There are many solutions to this equation. One solution of particular importance is

$E\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right),$

where A is the amplitude of the electric field, k is the wave number, and $\omega$ is the angular frequency. Combing this equation with [link] gives

${k}^{2}=\frac{{\omega }^{2}}{{c}^{2}}.$

According to de Broglie’s equations, we have $p=\hslash k$ and $E=\hslash \omega$ . Substituting these equations in [link] gives

$p=\frac{E}{c},$

or

$E=pc.$

Therefore, according to Einstein’s general energy-momentum equation ( [link] ), [link] describes a particle with a zero rest mass. This is consistent with our knowledge of a photon.

This process can be reversed. We can begin with the energy-momentum equation of a particle and then ask what wave equation corresponds to it. The energy-momentum equation of a nonrelativistic particle in one dimension is

$E=\frac{{p}^{2}}{2m}+U\left(x,t\right),$

where p is the momentum, m is the mass, and U is the potential energy of the particle. The wave equation that goes with it turns out to be a key equation in quantum mechanics, called Schrӧdinger’s time-dependent equation    .

## The schrӧdinger time-dependent equation

The equation describing the energy and momentum of a wave function is known as the Schrӧdinger equation:

$-\frac{{\hslash }^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}\text{Ψ}\left(x,t\right)}{\partial {x}^{2}}+U\left(x,t\right)\text{Ψ}\left(x,t\right)=i\hslash \frac{\partial \text{Ψ}\left(x,t\right)}{\partial t}.$

As described in Potential Energy and Conservation of Energy , the force on the particle described by this equation is given by

$F=-\frac{\partial U\left(x,t\right)}{\partial x}.$

This equation plays a role in quantum mechanics similar to Newton’s second law in classical mechanics. Once the potential energy of a particle is specified—or, equivalently, once the force on the particle is specified—we can solve this differential equation for the wave function. The solution to Newton’s second law equation (also a differential equation) in one dimension is a function x ( t ) that specifies where an object is at any time t . The solution to Schrӧdinger’s time-dependent equation provides a tool—the wave function—that can be used to determine where the particle is likely to be. This equation can be also written in two or three dimensions. Solving Schrӧdinger’s time-dependent equation often requires the aid of a computer.

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there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
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