I'd like some help with something in this introduction to second quantization ... http://yclept.ucdavis.edu/course/242/Class.html(adsbygoogle = window.adsbygoogle || []).push({});

They start by considering two operators ##a## and ##a^\dagger## such that ##[a,a^\dagger]=1## and they are adjoint to each other. Also introduced is a state ##|\alpha\rangle## that is an eigenvector of the (Hermitian) operator ##a^\dagger a##, so that

##a^\dagger a |\alpha\rangle = \alpha |\alpha\rangle##.

They go on to show that the state ##a|\alpha\rangle## is an eigenstate of ##a^\dagger a## with eignvalue ##\alpha-1##, and ##a^\dagger|\alpha\rangle## is an eigenstate of ##a^\dagger a## with eigenvalue ##\alpha-1##.

Up to this point I managed to follow the plot reasonably well.

But now they say that the above"implies"that

##|\alpha-1\rangle=\frac 1{\sqrt\alpha}\alpha|\alpha\rangle##

and

##|\alpha+1\rangle=\frac 1{\sqrt{\alpha+1}}\alpha|\alpha\rangle##

which I am absulutely unable to understand. How do they get this result? where do the square roots come from? Why is there an asymmetry in the square roots?

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# I Second quantization and creation/annih. operators

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