Example of a linear combination, anyone have any insights?

[Edward Hunia]

According to A Borobantu, regarding superposition, where Ψ is a state vector or a wave amplitude, given Ψ₁, Ψ₃, Ψ₃... Ψ_ncan by physically realized, then the following statement holds for all linear combinations

Ψ = ∑c_iΨ_i

where c_i can satisfy 1 = ∑|c_k|² to be called a complete and closed system.

then a generalized linear statement is
Ψ =e^iθ₁Ψ₁ + e^iθ₂Ψ₂ +...e^iθ_nΨ_n

any input appreciated please. what ever insights you may have would be greatly appreciated please.

 

[bhobba]

According to A Borobantu, regarding superposition, where Ψ is a state vector or a wave amplitude, given Ψ₁, Ψ₃, Ψ₃... Ψ_ncan by physically realized, then the following statement holds for all linear combinations.

That's just the principle of superposition which, basically, says pure quantum states form a complex vector space.

Have you studied linear algebra?

If not I strongly suggest you become antiquated with it before delving deeper into QM:
https://www.math.ucdavis.edu/~linear/

Once you have that out of the way then its easy to understand the Dirac notation:
http://www.fysik.su.se/~edsjo/teaching/kvant2/pdf/formalism.pdf

where c_i can satisfy 1 = ∑|c_k|² to be called a complete and closed system. then a generalized linear statement is
Ψ =e^iθ₁Ψ₁ + e^iθ₂Ψ₂ +...e^iθ_nΨ_n

The first bit contradicts the second - the square of absolute values of the coefficients do not sum to one ie are not normalised. However when you know a bit of linear algebra sorting out what's going on should be easy.

That said if you are starting out in QM I personally wouldn't be using books that require that level of mathematical background - I would start with something a lot friendlier like Susskinds book:
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

Thanks
Bill

 

[Edward Hunia]

That's just the principle of superposition which, basically, says pure quantum states form a complex vector space.

Have you studied linear algebra?

If not I strongly suggest you become antiquated with it before delving deeper into QM:
https://www.math.ucdavis.edu/~linear/

Once you have that out of the way then its easy to understand the Dirac notation:
http://www.fysik.su.se/~edsjo/teaching/kvant2/pdf/formalism.pdf
The first bit contradicts the second - the square of absolute values of the coefficients do not sum to one ie are not normalised. However when you know a bit of linear algebra sorting out what's going on should be easy.

That said if you are starting out in QM I personally wouldn't be using books that require that level of mathematical background - I would start with something a lot friendlier like Susskinds book:
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

Thanks
Bill

Hi Bill. It's not a contradiction, its a constraint on the linear system that when satisfied classifies the system as closed. If the system is physically realizable, and it satisfies the constraint, then the system is said to be complete and closed. Do you know what that means?

Thanks for all of the links also. The math is straight forward enough, unless you see something wrong, then please point it our!

 

[bhobba]

Hi Bill. It's not a contradiction, its a constraint on the linear system that when satisfied classifies the system as closed. If the system is physically realizable, and it satisfies the constraint, then the system is said to be complete and closed. Do you know what that means?

I have zero idea what they are on about - closed etc is not terminology I have heard in relation to this stuff - and I have read and studied a LOT of books on QM. Closed is part of the definition of a Hilbert space which has a technical meaning not really related to this. Physically realizable is usually related to the advanced Rigged Hilbert Space approach which is definitely nor recommended for the beginner.

I tried a search on A Borobantu and couldn't find anything. The nearest I could find is:
http://arxiv.org/abs/physics/0602145

Personally I would not recommend learning QM from that. If you want an axiomatic approach I think the following is better:
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20

Even more elegant, but more advanced mathematically is a post I did a while ago now (see post 137):
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Thanks for all of the links also. The math is straight forward enough, unless you see something wrong, then please point it our!

Like I said its using terminology I have never seen before, and the so called generalised linear statement makes no sense at all - it most certainly is not a generalised superposition.

Thanks
Bill

 

[Edward Hunia]

I have zero idea what they are on about - closed etc is not terminology I have heard in relation to this stuff - and I have read and studied a LOT of books on QM. Closed is part of the definition of a Hilbert space which has a technical meaning not really related to this. Physically realizable is usually related to the advanced Rigged Hilbert Space approach which is definitely nor recommended for the beginner.

I tried a search on A Borobantu and couldn't find anything. The nearest I could find is:
http://arxiv.org/abs/physics/0602145

Personally I would not recommend learning QM from that. If you want an axiomatic approach I think the following is better:
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20

Even more elegant, but more advanced mathematically is a post I did a while ago now (see post 137):
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7
Like I said its using terminology I have never seen before, and the so called generalised linear statement makes no sense at all - it most certainly is not a generalised superposition.

Thanks
Bill

Hi Bill,
I guess it's wrong to think that cosθ_i + isinθ_i is a complex operation that supplies linearity... but would help if you show why? lol

My math exploration lead me to the definition of Cauchy sequences; which I studied at uni, but had forgotten. There in I gained the understanding of completeness... its essential meaning confines the entire space to real terms for every physical point. From that it seems logical that having a modulus of one allows the functional mapping to be completely spanned within such a defined space. hence complete and closed.

I'm having trouble with notation myself. Interesting too, the precise definition of postulate 1 throws me due to the distinction of state and time; when it comes to dirac notation bra-ket because a state vector can be fully describe without a time variable. From there, time denotes the evolution of the state. bra ket notation presupposes time evolution. so I'm not sure how to represent the probability density in terms of a bra ket? i.e. Ψ_(a,b) = <b|a> so |Ψ_(a,b)|² = ?

From what I could gather, it is averaged out, and that average is the conjugate (re complex Hilbert space); its been a while since I studied Hilbert space.

So I'm not sure how to represent a bra ket probability density, or if its even sensible to consider such an idea (seems like it should be ok due to Kronicker delta simplifications). And Wikipedia didn't allow me to easily connect the dots... at least not yet.

 

[DrClaude]

The OP is gone.

Thread closed.