Ehrenfest theorem

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

Mathematically, this is expressed in its most general form such that the complete time derivative of the expectation value of a quantum mechanical operator is related to the commutator of this operator and the Hamilton operator as follows: ${\displaystyle H}$

${\displaystyle {\frac {d}{dt}}\langle O\rangle ={\frac {i}{\hbar }}\langle [H,O]\rangle +\left\langle {\frac {\partial O}{\partial t}}\right\rangle }$

It represents a quantum mechanical operator and its expected value . ${\displaystyle O}$ ${\displaystyle \langle O\rangle }$

Classic analogue

In the Hamilton formalism of classical mechanics, the following applies to the time evolution of a phase space function:

${\displaystyle {\frac {d}{dt}}f(p,q,t)=-\{H,f\}+{\frac {\partial f}{\partial t}}}$

with the Poisson bracket . When quantizing , the Poisson bracket is replaced by the commutator multiplied by. The quantum mechanical analog of a phase space function is an operator ( observable ). Thus the Ehrenfest theorem is the direct analogue of the above classical statement. ${\displaystyle \{H,f\}=\nabla _{q}H\nabla _{p}f-\nabla _{p}H\nabla _{q}f}$ ${\displaystyle {\tfrac {1}{i\hbar }}}$

Derivation

The following derivation uses the Schrödinger picture . For an alternative consideration in the Heisenberg picture, see "Equation of motion for expected values" under Heisenberg's equation of motion .

Let the system under consideration be in the quantum state . One thus obtains for the time derivative of the expectation value of an operator O: ${\displaystyle \Psi }$

${\displaystyle {\begin{aligned}{\frac {d}{dt}}\langle O\rangle &={\frac {d}{dt}}\int \Psi ^{*}O\Psi dV=\int \left[\left({\frac {\partial \Psi ^{*}}{\partial t}}\right)O\Psi +\Psi ^{*}\left({\frac {\partial O}{\partial t}}\right)\Psi +\Psi ^{*}O\left({\frac {\partial \Psi }{\partial t}}\right)\right]dV\\&=\int \left[\left({\frac {\partial \Psi ^{*}}{\partial t}}\right)O\Psi +\Psi ^{*}O\left({\frac {\partial \Psi }{\partial t}}\right)\right]dV+\left\langle {\frac {\partial O}{\partial t}}\right\rangle \end{aligned}}}$

Now consider the Schrödinger equation

${\displaystyle {\frac {\partial \Psi }{\partial t}}=-{\frac {i}{\hbar }}H\Psi }$

If we conjugate this equation and note that the Hamilton operator is self-adjoint , it follows ${\displaystyle H}$

${\displaystyle {\frac {\partial \Psi ^{*}}{\partial t}}={\frac {i}{\hbar }}\Psi ^{*}H}$.

Inserting these relations yields:

${\displaystyle {\frac {d}{dt}}\langle O\rangle ={\frac {i}{\hbar }}\int [\Psi ^{*}HO\Psi -\Psi ^{*}OH\Psi ]dV+\left\langle {\frac {\partial O}{\partial t}}\right\rangle ={\frac {i}{\hbar }}\langle [H,O]\rangle +\left\langle {\frac {\partial O}{\partial t}}\right\rangle }$

application

Position and momentum operators

For the special case of the momentum operator (this is not explicitly time-dependent, that is ), according to the Ehrenfest theorem: ${\displaystyle {\frac {\partial p}{\partial t}}=0}$

${\displaystyle {\frac {d}{dt}}\langle p\rangle ={\frac {i}{\hbar }}\langle [H,p]\rangle ={\frac {i}{\hbar }}\langle [{\frac {p^{2}}{2m}}+V(x),p]\rangle ={\frac {i}{\hbar }}\langle [V(x),p]\rangle }$

Now the commutator is evaluated in the position representation, i.e. with , and : ${\displaystyle [V,p]}$ ${\displaystyle p=-i\hbar \nabla }$ ${\displaystyle V=V(x)}$ ${\displaystyle \Psi =\Psi (x)}$

${\displaystyle [V,-i\hbar \nabla ]\Psi =-i\hbar V\nabla \Psi -(-i\hbar \nabla (V\Psi ))=-i\hbar V\nabla \Psi +i\hbar (\nabla V)\Psi +i\hbar V(\nabla \Psi )=i\hbar (\nabla V)\Psi \quad \Rightarrow \quad [V,p]=i\hbar (\nabla V)}$

The time derivative of the expected impulse value in the position representation is therefore:

${\displaystyle {\frac {d}{dt}}\langle p\rangle ={\frac {i}{\hbar }}\langle i\hbar \nabla V(x)\rangle =-\langle \nabla V(x)\rangle }$

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

${\displaystyle {\frac {d}{dt}}\langle x\rangle ={\frac {i}{\hbar }}\langle [H,x]\rangle ={\frac {i}{\hbar }}\langle [{\frac {p^{2}}{2m}}+V(x),x]\rangle ={\frac {i}{\hbar }}{\frac {1}{2m}}\langle [p^{2},x]\rangle ={\frac {i}{\hbar }}{\frac {1}{2m}}\langle p\underbrace {[p,x]} _{-i\hbar }+\underbrace {[p,x]} _{-i\hbar }p\rangle ={\frac {1}{m}}\langle p\rangle }$

The simple commutator relation as well as the canonical commutation relations between momentum and position operator were used. ${\displaystyle [p^{2},x]=p[p,x]+[p,x]p}$

From the two derived relationships

${\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle \ ,\quad {\frac {d}{dt}}\langle p\rangle =-\langle \nabla V(x)\rangle }$

follows:

${\displaystyle m{\frac {d^{2}}{dt^{2}}}\langle x\rangle =-\langle \nabla V(x)\rangle =\langle F(x)\rangle }$

Here the force was used as a negative gradient of the potential. The equations of motion for the expected value of the position and momentum operator are almost identical to those of classical mechanics, but the force at the expected value of the location is replaced by the expected value of the force . If the force is not a linear function of the position, the expected value cannot be absorbed into the argument and classical and quantum mechanical equations of motion differ from each other. ${\displaystyle F(x)}$ ${\displaystyle F(\langle x\rangle )}$ ${\displaystyle \langle F(x)\rangle }$

Classic approximation

The expected value of the force can be expanded into a Taylor series around the expected value of : ${\displaystyle F(x)}$ ${\displaystyle x}$

${\displaystyle {\begin{aligned}\left\langle F(x)\right\rangle &=\left\langle F(\langle x\rangle )+F^{\prime }(\langle x\rangle )(x-\langle x\rangle )+{\frac {1}{2}}F^{\prime \prime }(\langle x\rangle )(x-\langle x\rangle )^{2}+{\mathcal {O}}(x^{3})\right\rangle \\&=F(\langle x\rangle )+F^{\prime }(\langle x\rangle )\underbrace {\left\langle (x-\langle x\rangle )\right\rangle } _{=0}+{\frac {1}{2}}F^{\prime \prime }(\langle x\rangle )\underbrace {\left\langle (x-\langle x\rangle )^{2}\right\rangle } _{=(\Delta x)^{2}}+\left\langle {\mathcal {O}}(x^{3})\right\rangle \\&=F(\langle x\rangle )+{\frac {1}{2}}F^{\prime \prime }(\langle x\rangle )(\Delta x)^{2}+\left\langle {\mathcal {O}}(x^{3})\right\rangle \end{aligned}}}$

If one only takes into account the first summand, one obtains

${\displaystyle \langle F(x)\rangle \approx F(\langle x\rangle )\qquad (*)}$

and thus

${\displaystyle m{\frac {d^{2}}{dt^{2}}}\langle x\rangle =F(\langle x\rangle )}$.

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

${\displaystyle m{\frac {d^{2}}{dt^{2}}}x=F(x)}$.

The assumption (*) and thus also the classical equation of motion for quantum mechanical expectation values only apply exactly if the force F (x) is a linear function of the position x. This applies to the simple cases of the harmonic oscillator or the free particle (then all local derivatives of the force of degree greater than or equal to 2 vanish). In addition, it can be said that (*) applies when the width of the probability of stay is small compared to the typical length scale on which the force F (x) varies.

The equation of motion for expected values ​​reads with the next non-zero correction to the classical equation of motion:

${\displaystyle m{\frac {d^{2}}{dt^{2}}}\langle x\rangle =F(\langle x\rangle )+{\frac {1}{2}}F^{\prime \prime }(\langle x\rangle )(\Delta x)^{2}}$

See also

• Liouville equation , Liouville theorem (physics)

literature

• Leslie E. Ballentine: Quantum Mechanics: A Modern Development . 1st edition. World Scientific Publishing, Singapore 1998, ISBN 981-02-4105-4
• P. Ehrenfest: Comment on the approximate validity of classical mechanics within quantum mechanics . Zeitschrift für Physik A Issue 45, Numbers 7–8 / July, 1927, pp. 455–457.