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I was wondering why creation and destruction operators a+ and a- do not conmute.

Of course, I can show that they don't conmute by computing the conmutator [a+, a-] = -1. But I want to know the "physical" meaning of this.

Isn't destruction/creation a symmetric transformation? We "go up the ladder" with a+ and we "go down the ladder" with a-. Shouldn't they therefore cancel each other, i.e. a+a-=a-a+ = I?

Mathematically [itex] a_- a_+ \vert n \rangle = a_-\vert n+1 \rangle = \vert n \rangle [/itex]

Thank you for your time.

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# Why don't creation and destruction operators conmute?

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