I am reading the book Introduction to Quantum Mechanics by David Griffiths and have come to the section on Dirac notation. It explains that the state of a quantum system is represented by a vector |β(t)> living out in Hilbert space, and, as with any vector, is independent of the choice of basis.(adsbygoogle = window.adsbygoogle || []).push({});

It then explains that the position-space-wave-fuction [itex]\Psi(x,t)[/itex] is the coefficient in the expansion of |β(t)> in the basis of position eigenfuctions:

[itex]\Psi(x,t)=<x|\beta(t)>[/itex] , where |x> is the eigenfunction of the position operator with eigenvalue x.

The momentum-space-wave-function [itex]\Phi(p,t)[/itex] is the coefficient in the expansion of |β(t)> in the basis of momentum eigenfunctions:

[itex]\Phi(p,t)=<p|\beta(t)>[/itex] , where |p> is the eigenfunction of the momentum operator with eigenvalue p.

And similarly you can expand |β(t)> in the basis of energy eigenfunctions:

[itex]c_{n}(t)=<n|\beta(t)>[/itex] , where |n> is the nth eigenfunction of the Hamiltonian (assumed discrete).

So |β(t)> can be written in different way depending on the basis you choose. But I have a few questions which I'm hoping someone can clear up for me:

1)What is |β(t)>? Am I correct in thinking that |β(t)> is [itex]\Psi(x,t)[/itex] when you're working in the position space, and |β(t)> is [itex]\Phi(p,t)[/itex] when you're working in the momentum space? So when I said at the start "|β(t)> lives out in Hilbert space", it's actually position Hilbert space or momentum Hilbert space, depending on which you're working with? And the different spaces will have different eigenfunctions for the operators?

2)I am a bit confused by the dimension of |β(t)>. The eigenvalues of the position operator are always continuous (I think?), but say for instance the energy spectrum is discrete (or perhaps there are only a finite number of allowed energies). Are there not many more position eigenfunctions than energy eigenfunctions? Then when we write |β(t)> in each of these basis, will the dimensions not be different?

I may have a few more questions but I'll leave it to those for the moment.

Thanks for any help!

Jon.

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# Representing the state of a quantum system in different bases

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