Keywords |
• Physics,
• Chemistry

# Correspondence principle

The Bohr correspondence principle states that at the limit of large quantum numbers for atomic systems the formulae of classical physics should be found. This principle has especially made it possible to connect the formulae giving the radiation intensities of quantum atoms to classical formulae. Therefore it has guided theoreticians toward an interpretation of quantum equations by connecting them to their classical limits.

More generally, the principle of correspondence in quantum mechanics gives the rules for constructing quantum equations from classical equations.

Thus, starting from the formalism of Hamilton analytical mechanics, with for example the Hamiltonian function H of N particles in a potential V:

$H= \sum_{i=1}^{3N} \frac {p_{i}}{2m_{i}}+ V(q_{i})$

where $q_{i}$ and $p_{i}$ are the generalised coordinates of position and momentum, we know that the equations of motion for the system will be given by the following Hamilton equations:

$\dot{p_{i}}=-\frac{\partial H}{\partial q_{i}}$
$\dot{q_{i}}=\frac{\partial H}{\partial p_{i}}$

the dot denotes a derivative with respect to time.

Furthermore, for functions of generalised coordinates such as $F(q_{i},p_{i})$, there will respectively be the Poisson brackets:

$[F , G]=\sum_{i=1}^{3N}\frac{\partial F}{\partial q_{i}} \frac{\partial G}{\partial p_{i}} - \frac{\partial F}{\partial p_{i}} \frac{\partial G}{\partial q_{i}}$

$[q_{i},p_{j}]=\delta_{ij}$

and the Poisson equations:

$\frac{dF}{dt}=[F,H]+\frac{\partial F}{\partial t}$

The correspondence rules stipulate that the observable values must be replaced by linear Hermitian operators $\hat A$ in the previous equations to obtain the quantum equations.

Thus the Poisson brackets will become

$[\hat x_j, \hat p_k]=i\hbar\delta_{jk}$

where

$\hat x_j \equiv x_j$

$\hat p_k = -i \hbar \frac{\partial }{\partial x_k}$

$\hat{\vec{p}}=-i\hbar \nabla$

In particular for a value $\hat A$ with a Hamiltonian operator $\hat H$, the following Heisenberg equation is obtained:

$\frac{d \hat A}{dt}=\frac{1}{i\hbar}[\hat A, \hat H]+\frac{\partial \hat A}{\partial t}$

Finally, the Hamilton-Jacobi equation

$H(q_k,p_k)=-\frac{\partial S}{\partial t}$

will be transformed into the Schrödinger equation for N particles in a potential with the Hamiltonian operator

$\hat H(\hat q_k,-i \hbar \frac{\partial }{\partial q_k})=i \hbar \frac{\partial }{\partial t}$

And

$\hat H \psi =i \hbar \frac{\partial \psi}{\partial t}$

where $\psi = \psi (q_1, q_2,....q_N)$ is a wave function in the configuration space for N particles.

### connexes

#### Definition

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