Ehrenfest-Theorem

The Ehrenfest theorem, named after the Austrian physicist Paul Ehrenfest, establishes a connection between classical mechanics and quantum mechanics in physics. It states that the classical equations of motion for the mean values ​​of quantum mechanics apply under certain conditions; therefore contain classical mechanics to some extent in quantum mechanics is (correspondence principle).

Mathematically expressed that in its most general form, so from that full time derivative of the expectation value of a quantum mechanical operator to the commutator of this operator and the Hamiltonian is related as follows:

It represents a quantum mechanical operator and its expectation value

Classic analogue

In the Hamiltonian formalism of classical mechanics is valid for the time evolution of a phase space function:

With the Poisson bracket. During quantization, the Poisson bracket is replaced by the commutator with multiplied. The quantum mechanical analog of a phase space function is an operator ( observable ). Thus, the Ehrenfest theorem is a direct analogue to the above classical statement.

Derivation

The following derivation uses the Schrödinger picture. For an alternative view in the Heisenberg picture, see " equation of motion for expectation values ​​" under Heisenberg equation of motion.

It is the considered system in the quantum state. One thus obtains for the time derivative of the expectation value of an operator O:

We now consider the Schrödinger equation

Conjugate of this equation, noting that the Hamiltonian is self-adjoint, it follows

Substituting these relations now provides:

Application

Position and momentum operators

For the special case of the momentum operator ( this is not explicitly time dependent, that is ) applies after the Ehrenfest theorem:

Now the commutator is evaluated in the coordinate representation, ie with, and:

The time derivative of the pulse in the expected value of the local representation is:

Since the position operator is not explicitly time dependent, follows with the Ehrenfest theorem for the time evolution:

The simple Kommutatorrelation and the canonical commutation between momentum and position operator were used.

From the two derived relations

Follows:

Here the force was used as the negative gradient of the potential. The expectation values ​​of the position and momentum operators satisfy so from Newtonian mechanics usual equations, where, however, we take the expected find the expression. This leads us to the so-called classical approximation.

Classical approximation

The expectation value of the force can be expanded in a Taylor series around the expected value of developing:

Considering only the first term, we obtain

And thus

In words, this means that moves the expectation value of the position on a classical path, ie the classical equation of motion follows. The Ehrenfest theorem thus leads directly to an analogy of quantum mechanics to classical mechanics - here in the form of Newton's second axiom

The assumption (* ), and hence also the classical equation of motion for quantum mechanical expectation values ​​are, however, only exact if the force F ( x ) is a linear function of the position x. This applies to the simple case of the harmonic oscillator or the free particle ( then disappear all spatial derivatives of the power of degree greater than 2). Also it can be said that (* ) is considered, if the width of the probability distribution is small relative to the typical length scale on which varies the force F (x).

The equation of motion for expectation values ​​is the next non-vanishing correction to the classical equation of motion: