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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 ,


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.

Questions & Answers

For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
acceleration also inc
which paper will be subjective and which one objective
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
this msg is out of mistake. sorry friends​.
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat Reply
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
you are correct
law of multiple
if we heated the ice then the refractive index be change from natural water
can someone explain normalization condition
Priyojit Reply
please tell
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
Practice Key Terms 3

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