Understanding the Difference Between |z>, |+z>, and |-z> in Quantum Mechanics

[Agnostic]

does |z> = |+z> + |-z> ?

 

[Logarythmic]

No, |+z> + |-z> = |z> - |z> = 0.

 

[Agnostic]

No, |+z> + |-z> = |z> - |z> = 0.

umm..., no.

 

[Hargoth]

What is $| z \rangle$?

 

[Agnostic]

What is $| z \rangle$?

In quantum mechanics, |z> is a state vector read at "ket z".

It describes the state a particle is in.

 

[Hargoth]

Yeah, but if $| z_+ \rangle, | z_- \rangle$ are basekets of the Hilbert space you consider, your equation would be a definition of $| z \rangle$

 

[Agnostic]

Yeah, but if $| z_+ \rangle, | z_- \rangle$ are basekets of the Hilbert space you consider, your equation would be a definition of $| z \rangle$

is it a valid/correct definition?

I'm in an intro quantum class and I need to calculate:


so far, we have just been calculating things like: <+or-phi|+or-psi>

Now we are asked to calculate things like:

<-z|x>

Which i read as that is the amplitude of something in either the +x or -x state being in the -z state.

 

[jonestr]

does |z> = |+z> + |-z> ?

No since |z>=(1,0) in the z basis and |-z>= (0,1) in the z basis you could a. never get a scalar under addition and you could not get an answer of the zero vector since these vectors are linearly independent and form a complete basis. For your previous post you need to calculate what |-z> is in the x basis or what |x> is in the z basis to compute the inner product. Griffiths QM or Liboff are good sources for this. As is Nielsen and Chuang

Hope that helps

 

[Agnostic]

No since |z>=(1,0) in the z basis and |-z>= (0,1) in the z basis you could a. never get a scalar under addition and you could not get an answer of the zero vector since these vectors are linearly independent and form a complete basis. For your previous post you need to calculate what |-z> is in the x basis or what |x> is in the z basis to compute the inner product. Griffiths QM or Liboff are good sources for this. As is Nielsen and Chuang

Hope that helps

|z> is not equal to |+z>

I thought |+z>=(1,0)

 

[Hargoth]

is it a valid/correct definition?

I'm in an intro quantum class and I need to calculate:


so far, we have just been calculating things like: <+or-phi|+or-psi>

Now we are asked to calculate things like:

<-z|x>

Which i read as that is the amplitude of something in either the +x or -x state being in the -z state.

For a QM-Interpretation you have to normalize the statevector, so that
$\langle z | z \rangle = 1$. If $\langle z_+ | z_+ \rangle = 1$ and $\langle z_- | z_- \rangle = 1$-, this not the case here.

I wouldn't say "amplitude" but "probability": $|\langle -z | x \rangle|^2$ is the probability to measure "z-spin-down" on a particle of which you know it is in state "x-spin-up".

 

[Agnostic]

For a QM-Interpretation you have to normalize the statevector, so that
$\langle z | z \rangle = 1$. If $\langle z_+ | z_+ \rangle = 1$ and $\langle z_- | z_- \rangle = 1$-, this not the case here.

I wouldn't say "amplitude" but "probability": $|\langle -z | x \rangle|^2$ is the probability to measure "z-spin-down" on a particle of which you know it is in state "x-spin-up".

<-z|-z> = 1

<-z|-z> means what is the "probability" amplitude that a particle in state |-z> will be in state |-z>

 

[Hargoth]

Yeah, I just wanted to say that your probability of finding z in state z from the equation above would be $2^2=4$, so you have to normalize.

 

[jonestr]

you don't have to write the plus explicity. |z>=|+z>=(1,0) which is not equal to -|-z>=(0,-1). The negatives are part of the nomenclature and do not have algebraic significance.