Schroedinger Equation from Variational Principle

In summary, Landau's nonrelativistic quantum mechanics uses a variational principle to derive Schroedinger's equation, which can be obtained by varying both ##\psi## and ##\psi^{\ast}## independently. This allows for the determination of allowed energies by minimizing the difference between the expectation value of the Hamiltonian and the energy. This method is analogous to classical mechanics and can be used to compute observables through a path integral.
  • #1
Sturk200
168
17
Landau's nonrelativistic quantum mechanics has a "derivation" of Schroedinger's equation using what he calls "the variational principle". Apparently such a principle implies that:

$$\delta \int \psi^{\ast} (\hat{H} - E) \psi dq = 0$$

From here I can see that varying ##\psi## and ##\psi^{\ast}## independently gives rise to the equation ##\hat{H} \psi = E \psi##. But where does that first equation come from? I think it is saying that the the allowed energies are those for which the difference between the expectation value of the hamiltonian and the energy is minimized. I'm not sure why that should be so. Is there some analogue to classical mechanics that I'm missing?

(Note: this is section 20 of the book).
 
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  • #2
If you have a classically motivated differential equation of motion for some degree(s) of freedom, you can usually hack your way towards a Lagrangian that yields that equation. A reasonable starting point is to multiply the equation on the left by your DOF (here, [itex]\psi[/itex]) and start integrating by parts.

In classical mechanics this would usually be pretty pointless. In quantum theory this action can be used to construct a path integral, providing a different way of computing observables.
 

FAQ: Schroedinger Equation from Variational Principle

1. What is the Schrödinger Equation?

The Schrödinger Equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum state. It was first introduced by Austrian physicist Erwin Schrödinger in 1926.

2. What is the Variational Principle?

The Variational Principle is a mathematical principle that is used to find the most accurate description of a physical system. It states that the true physical state of a system is the one that minimizes the expectation value of the energy.

3. How is the Schrödinger Equation derived from the Variational Principle?

The Schrödinger Equation can be derived from the Variational Principle by considering a trial wavefunction, which is an approximation to the true physical state. By applying the Variational Principle, the trial wavefunction can be adjusted until the energy expectation value is minimized, resulting in the correct form of the Schrödinger Equation.

4. What is the significance of the Schrödinger Equation from the Variational Principle?

The Schrödinger Equation from the Variational Principle is significant because it provides a rigorous mathematical framework for describing the behavior of quantum systems. It allows us to make predictions about the behavior of particles at the microscopic level and has applications in a wide range of fields, including chemistry, materials science, and engineering.

5. What are some limitations of the Schrödinger Equation from the Variational Principle?

One limitation of the Schrödinger Equation from the Variational Principle is that it cannot be used to describe systems with multiple particles. In these cases, more complex equations, such as the Hartree-Fock method, must be used. Additionally, the Schrödinger Equation only provides a probabilistic description of a system, and cannot give definite predictions for the behavior of individual particles.

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