hmparticle9
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The following sentence appears in my book (Introduction to Quantum Mechanics by Griffiths (3rd Edition)) :
"## \langle x | p \rangle ## is the momentum eigenstate (with eigenvalue p) in the position basis." Page 122.
I am not sure why this is the case. I have done some digging... on a previous page (114) it says
"The momentum space wave function ##\Phi(p ,t)## is the ##p## component in the expansion of ## | S(t) \rangle ## in the basis of momentum eigenfunctions:
$$\Phi(p ,t) = \langle p | S(t) \rangle$$
(with ## | p \rangle ## standing for the eigenfunction of ##\hat{p}## with eigenvalue ##p##)"
So is it convention that ## | p \rangle ## is the eigenfunction/eigenstate of ##\hat{p}##? If that is the case then that is one problem solved.
The second problem is, why does application of ## \langle x|## to ## | p \rangle ## mean that we are taking ##| p \rangle## with respect to the position basis?
From doing some further reading it seems as though that my second question could have the same answer as the first.
"## \langle x | p \rangle ## is the momentum eigenstate (with eigenvalue p) in the position basis." Page 122.
I am not sure why this is the case. I have done some digging... on a previous page (114) it says
"The momentum space wave function ##\Phi(p ,t)## is the ##p## component in the expansion of ## | S(t) \rangle ## in the basis of momentum eigenfunctions:
$$\Phi(p ,t) = \langle p | S(t) \rangle$$
(with ## | p \rangle ## standing for the eigenfunction of ##\hat{p}## with eigenvalue ##p##)"
So is it convention that ## | p \rangle ## is the eigenfunction/eigenstate of ##\hat{p}##? If that is the case then that is one problem solved.
The second problem is, why does application of ## \langle x|## to ## | p \rangle ## mean that we are taking ##| p \rangle## with respect to the position basis?
From doing some further reading it seems as though that my second question could have the same answer as the first.
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