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## Main Question or Discussion Point

I'd like some help with something in this introduction to second quantization ... http://yclept.ucdavis.edu/course/242/Class.html

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

##|\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?

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?