What is the physical interpretation of bra-ket notation in quantum mechanics?

[sampo]

Hi, I some basic questions on the physical meaning of bra-ket notation. I have looked at a lot of material, and I have seen descriptions of the algebraic properties of bra-kets, and I have seen hints at it having meaning regarding the probability of events happening/state changes, but I can't seem to find an exact depiction of this relationship. Hopefully this will be an easy one. If anyone could just let me know which, if any, of my statements are true, it would be most appreciated. Thanks, and feel free to elaborate.

 

[grzz]

There is a little book 'The Quantum World' by Polkinghorn. Perhaps you find it a good introduction.

 

[tom.stoer]

A ket is an abtract notation for a quantum state. You can project it on a bra-basis to get (e.g.) a wave function in position or momentum space

\psi(x) = \langle x|\psi\rangle

There are some spezial kets (special basis) with a physical interpretation; |x> is a position-eigenket, |p> is a momentum-eigenket, |E> is an energy-eigenket of a Hamiltonian H

For the wave functions you get

\psi_y(x) = \langle x|y \rangle = \delta(x-y)

\psi_p(x) = \langle x|p \rangle = e^{ipx}

i.e.a delta-function as a position-eigenfunction in position space and a plane-wave as a momentum-eigenfunction in position space, respectively.

 

[grzz]

Here is an analogy:

A vector v can be represented by its components in some coordinate basis.

e.g. v = ivx + jvy +kvz

Similarly a quantum state ψ can be represented by a ket |ψ> which can be expanded in terms of its components in some basis |k>.

i.e. |ψ> = Σ|k><k|ψ>

 

[sampo]

Okay, I did a little more research and found something, which I hope to be true. Lemme revise my earlier question according to the attached.

 

[Fredrik]

Regarding your questions about time:

No, there's isn't one Hilbert space for each value of t, or one inner product for each value of t. There's one Hilbert space that contains all the state vectors, and it's equipped with an inner product. I suggest that you think of it this way: Define \psi_t(\vec x)=\psi(\vec x,t). For each \vec x, \psi_t:\mathbb R^3\rightarrow\mathbb C is a state vector. The function t\mapsto\psi_t is a curve in the Hilbert space of state vectors. If we define \psi such that each \psi_t is normalized, it's actually a curve on the unit sphere in that Hilbert space. The Schrödinger equation tells you how to find that curve, given a single point on the curve.

There's a minor inaccuracy in what I just said. The "inner product" on the set of square-integrable functions is actually a "semi-inner product". See posts #4 and #6 here if you care about those details.

 

[sampo]

Okay, cool, this is great.

 

[sampo]

Oh, sorry I didn't mention it, but the pic on my last reply contains further questions if anyone can answer them. Especially the one at the end. Thanks.

 

[nigelscott]

Re: Reply #3 <x|y> and <x|p> are interpreted as inner products correct? Thus, I assume these can expanded in terms of integrals involving delta functions. <x|p> is easy to figure out but I don't understand how to get δ(x-y) for <x|y>.

 

[bhobba]

Re: Reply #3 <x|y> and <x|p> are interpreted as inner products correct? Thus, I assume these can expanded in terms of integrals involving delta functions. <x|p> is easy to figure out but I don't understand how to get δ(x-y) for <x|y>.

You have hit upon something actually quite difficult here - the use of Delta Functions in QM. Basically they do not belong to a Hilbert Space but to something called a Rigged Hilbert Space and coming to grips with that is a whole other story - you need to be very adept at analysis and delve into tombs like Gelfand and Vilenkin - Generalized Functions - tough going even for guys like me into that sort of stuff - but it must be said not impossible if you persevere.

At the beginner level however its best to simply view Delta Functions as an ordinary function that for all practical purposes behave like a delta function - but really isnt. Under that view <x|y> = integral delta (x-x') delta (y-x') dx' = delta (x-y).

Basically the idea of a Rigged Hilbert Space is this. In a Hilbert space H bras and kets can be put into one to one correspondence ie its vectors and the linear functions defined on those vectors - called its dual -can be put in one to one correspondence. But if instead of a Hilbert space (which consists of all square integrable functions) you consider the space of all test functions (which is a subset of square integrable functions) then its dual T* (ie the space of all linear functions defined on that space) is much larger than H and includes stuff like the delta function. So what you have is T subset of H subset of T* - which is called a Gelfland Triple and the basis of Rigged Hilbert Spaces.

For a bit of an introduction to Rigged Hilbert Spaces check out (which also explains what a test function is if you do not know it):
http://en.wikipedia.org/wiki/Distribution_(mathematics )
http://www.abhidg.net/RHSclassreport.pdf

As to its physical meaning that is contained in a theorem you won't find mentioned that often in textbooks but IMHO is very fundamental - Gleason's Theorem which basically says Born's rule is the only way you can define probabilities on a Hilbert Space.
http://en.wikipedia.org/wiki/Gleason's_theorem

In fact from the assumption that QM is theory whose pure states are complex vectors all of QM basically follows. Check out:
http://arxiv.org/pdf/quant-ph/0111068v1.pdf

Its slightly different to the way I mentioned (from Gleasons Theorem) but it gives you the gist.

Thanks
Bill

 

[nigelscott]

Thank you very much for your response. I had done a little research prior to my post and concluded that this was indeed a complicated issue. I think I will stick with the beginner view for my purposes, given that I am re-learning a lot of this stuff 25 years after leaving college! Thanks again.