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[parent] example of quantum commutator algebra (Example)

Here we illustrate a simple example of quantum commutator algebra using a one-dimensional quantum system. Let f(q) be a function of q. The three commutators of q and of each of the functions p2f(q), pf(q)p, and f(q)p2 may all be identified (to within the factor iℏ) with the derivative with respect to p of these functions, but they are not the same operators. Indeed, by repeated application of the commutator algebra rule

[q,G (p ,...,p )] = iℏ∂G-- i 1 R ∂pi
(1)

we get

2 [q,p f(q)] = 2iℏpf(q)

[q,pfp] = iℏ(fp + pf)

2 [q,fp ] = 2iℏfp

In the same way

ℏ [p,p2f] = -p2f ′ i

ℏ- ′ [p,pf p] = ipf p

2 ℏ- ′2 [p,fp ] = if p

0.1 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].


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Cross-references: work, domain, volume, quantum mechanics, commutator algebra, operators, commutators, function, system

This is version 2 of example of quantum commutator algebra, born on 2010-02-14, modified 2010-02-14.
Object id is 835, canonical name is ExampleOfQuantumCommutatorAlgebra.
Accessed 1775 times total.

Classification:
Physics Classification: 03.65.Ca (Formalism)
 03.65.Fd (Algebraic methods )
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