Understanding Spin States in Hilbert Space

[member 545369]

Hello

In our Quantum Mechanics lecture we have been discussing a simplified model of the Stern-Gerlach experiment. Let $|+>$ and $|->$ denote an electron that is "spin up" and "spin down" (with respect to $\hat{z}$), respectively. Our professor then asserted that $|+>$ and $|->$ acted as a basis for a 2-D Hilbert space. It follows that a spin in $\hat{x}$ could then be constructed in the following way: $$|+_x > = \frac{1}{\sqrt{2}} ( |+> + |->)$$ however, this confuses me. How can we possible construct a ket pointing in $\hat{x}$ out of kets pointing in $\hat{z}$??

 

[phyzguy]

Of course it is confusing. You have no experience with quantum mechanics because these things are not apparent at macroscopic scales. One way to say it is that, since the dimension of the Hilbert space is 2, any pair of kets which are linearly independent can act as a basis. So there has to be a way to express |+x> in terms of |+z> and |-z> since |+z> and |-z> span the whole space. This means that a particle that is in |+z> has some probability (it turns out to be 50%) of being in |+x> and some probability of being in |-x> (again 50%). I would recommend (as I often do) the Feynman lectures on Physics, Vol3 Ch 6, which explains this in some detail in a way which is fairly easy to understand.

 

[member 545369]

Thanks for the reply. So I guess my question now becomes: is there any form of mathematical justification for claiming that $|+>$ and $|->$ are linearly independent? Why isn't it the case that $|+> = (-1)|->$, just like how $\hat{z} = (-1) \hat{-z}$?

 

[PeroK]

Thanks for the reply. So I guess my question now becomes: is there any form of mathematical justification for claiming that $|+>$ and $|->$ are linearly independent? Why isn't it the case that $|+> = (-1)|->$, just like how $\hat{z} = (-1) \hat{-z}$?

These are spin states, not spin in a 3D direction. That's an important thing to start to understand. States exist in a complex Hilbert space, not in 3D real vector space space.

The Hilbert space of electron spin is only 2D, but that has no direct relationship to 2D or 3D physical space.

 

[member 545369]

States exist in a complex Hilbert space, not in 3D real vector space space.

That's so weird! I am very curious as to how a Hilbert Space describes states as "vectors". Unfortunately my Quantum book (McIntyre) is not delving much into the mathematics behind Quantum Mechanics…

Can anyone suggest a good text that isn't afraid to get into the mathematics behind Hilbert Spaces? (relative background: I've taken Analysis, complex variables, ODEs, PDEs, … etc)

 

[PeroK]

That's so weird! I am very curious as to how a Hilbert Space describes states as "vectors". Unfortunately my Quantum book (McIntyre) is not delving much into the mathematics behind Quantum Mechanics…

Can anyone suggest a good text that isn't afraid to get into the mathematics behind Hilbert Spaces? (relative background: I've taken Analysis, complex variables, ODEs, PDEs, … etc)

Well, the mathematics of spin-1/2 particles is about as simple as it gets: states are 2D complex vectors and operators are 2x2 complex matrices. The important point is that the state (vector/ket) contains all the information about the state.

In this case, the z-spin-up state is not just a certain spin in the z-direction, but tells you everything about any spin measurement. The state of z-spin-up means that a measurement of spin in the z-direction returns spin-up with 100%, which might suggest that it is simply spin-up in the z-direction. But, it also tells you that a measurement of spin in the x or y directions returns spin-up and spin-down with 50% each. And, using linear algebra, you can calculate the probabilities for measurements about any other axis as well.

I'm not familiar with McIntyre, but he ought to cover the maths as well as any other author. I think the book has a good reputation.

 

[phyzguy]

If using the word "vector" to describe the elements of a Hilbert space confuses you, you could call them "elements" or "states". As PeroK says, they are separate entities from vectors in 3D space. Although bear in mind that the 3D physical space of Euclidean geometry is also a Hilbert space.

 

[Jilang]

You might have less consternation if you considered polarisation. The |H> and |V> states are clearly orthogonal and any other vector might be built from a sum of the two. However with spin the states of up and down states are orthogonal so it’s more tricky to picture.