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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

2 E x 2 = 1 c 2 2 E t 2 ,

where c is the speed of light and the symbol 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 ( x , t ) | 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 ( x , t ) = A sin ( k x ω t ) ,

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

k 2 = ω 2 c 2 .

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

p = E c ,

or

E = p c .

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 = p 2 2 m + U ( x , t ) ,

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:

2 2 m 2 Ψ ( x , t ) x 2 + U ( x , t ) Ψ ( x , t ) = i Ψ ( x , t ) t .

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

F = U ( x , t ) 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.

Practice Key Terms 3

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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