Physics Library
 An open source physics library
Encyclopedia | Forums | Docs | Random | Template Test |  
Login
create new user
Username:
Password:
forget your password?
Main Menu
Sections

Talkback

Downloads

Information
[parent] examples of constants of the motion (Example)

There exists an observable which always commutes with the hamiltonian: the Hamiltonian itself. The energy is therefore a constant of the motion of all systems whose Hamiltonian does not depend explicitly upon the time.

As another possible constant of the motion, let us mention parity. We denote under the name of parity the observable $P$ defined by

$\displaystyle P \psi(q) = \psi(-q)$ (1)

It is easily verified that $P$ is Hermitean. Moreover, $P^2=1$ and, consequently, the only possible eigenvalues of $P$ are $+1$ and $-1$; even functions are associated with $+1$, and odd functions with $-1$.

When the Hamiltonian is invariant under the substitution of $-q$ for $q$, we obviously have

$\displaystyle [P,H] = 0$

Indeed, if

$\displaystyle H\left(\frac{\hbar}{i} \frac{d}{dq},q\right) = H\left(-\frac{\hbar}{i} \frac{d}{dq},-q\right) $

one has, for any $\psi(q)$,

$\displaystyle PH\psi = H\left(-\frac{\hbar}{i} \frac{d}{dq},-q\right)\psi(-q)=H\left(\frac{\hbar}{i} \frac{d}{dq},q\right)\psi(-q) = HP\psi$

Under these conditions, if the wave function has a definite parity at a given initial instant of time, it conserves the same parity in the course of time.

This property is easily extended to a system having an arbitrary number of dimensions; in particular, it applies to systems of particles for which the parity operation amounts to a reflection in space $(\mathbf{r}_i \rightarrow -\mathbf{r}_i)$ and for which the observable parity is defined by

$\displaystyle P\Psi(\mathbf{r}_1,\mathbf{r}_2.\dots)=\Psi(-\mathbf{r}_1,-\mathbf{r}_2,\dots)$

References

[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public domain work [1].



"examples of constants of the motion" is owned by bloftin.

View style:

Also defines:  parity

This object's parent.

Cross-references: work, domain, volume, quantum mechanics, operation, wave, functions, systems, motion, energy, Hamiltonian, commutes, observable
There are 3 references to this object.

This is version 1 of examples of constants of the motion, born on 2010-02-14.
Object id is 837, canonical name is ExamplesOfConstantsOfTheMotion.
Accessed 541 times total.

Classification:
Physics Classification: 03.65.Ca (Formalism)
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:

No messages.

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "