## Angular Momentum Operators and Commutation Relations in Quantum Physics

In is one of the fundamental differences of quantum physics with classical mechanics that in the quantum world we cannot know all measurements to absolute precision. Some quantities 'pair up' so absolute knowledge of one precludes absolute knowledge of the other.

Perfect knowledge of a particles momentum means its position cannot be known with precision.

In fact we can say the product of the uncertainty in the position and the uncertainty in the momentum of a particle must be greater than or equal to

Particles created from the vacuum must have a certain nonzero energy. We can treat this energy as an uncertainty because it is energy created almost from nothing. Knowledge of this energy puts an upper limit on the lifetime of the particle.

In fact we can say the product of the reciprocal of uncertainty of the energy of the particle and the reciprocal in the lifetime of the particle must be greater than or equal to

We cannot know simultaneously the angular momentum of a particle around two different axes. The product of the uncertainties must be at least

Each of these three examples gives an example of two quantities that cannot be simultaneously known to arbitrarily high precision. For each pair, this is the case because their quantum mechanical operators do not commute.

For exampleand

Dropping thegives the result

Similar relationships hold for every pair of quantities that cannot be written down simultaneously to arbitrarily high precision.

If two quantum mechanical operators hat x and hat b do commute thenand the two physical quantities corresponding to these two quantum mechanical operators can simultaneously be know to arbitrarily high position,