- #1
demoncore
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Sorry for disregarding the template; I'm not really working out a homework problem as much as just trying to follow the reasoning in the text. I'm working through the first chapter of Quantum Mechanics, McIntyre, and I'm a little bit confused by the following.
The text introduces bra-ket notation in the context of the Stern-Gerlach experiment, deriving some results having to do with the spin measured along various axes and relating it to the state vector. (Please read the _ sign as a subscript)
They write the 'general form' of the S_x state kets in terms of the S-z bases kets:
|+>_x = a |+> + b |->
|->_x = c |+> + d |->,
They also have the following experimental results:
|<+|+>|^2 = |<-|+>|^2 = |<+|->|^2 = |<-|->^2 = 0.5 (all S_x state vectors)
expanding and equating they find that |a|^2 = |b|^2 = |c|^2 = |d|^2 = 0.5
This is the paragraph that confuses me--I think it might be my unfamiliarity with complex operations:
Because each coefficient is complex, each has an amplitude and phase. However, the overall phase of a quantum state vector is not physically meaningful. Only the relative phase between different components of the state vector is physically measurable. Hence, we are free to choose one coefficient of each vector to be real and positive without any loss of generality. This allows us to write the desired states as:
|+>_x = (1/√2) [ |+> + e^(iα) |->]
|->_x = (1/√2) [ |+> + e^(iβ) |->]
I am a little confused by the reasoning here. I suppose |a|^2 = (1/√2) => a = (1/√2) * e^(iθ); the absolute phase of the state vector doesn't matter, so we are free to take θ= 0 for one coefficient of each vector?
Please correct any terminology/reasoning errors I have made here.
The text introduces bra-ket notation in the context of the Stern-Gerlach experiment, deriving some results having to do with the spin measured along various axes and relating it to the state vector. (Please read the _ sign as a subscript)
They write the 'general form' of the S_x state kets in terms of the S-z bases kets:
|+>_x = a |+> + b |->
|->_x = c |+> + d |->,
They also have the following experimental results:
|<+|+>|^2 = |<-|+>|^2 = |<+|->|^2 = |<-|->^2 = 0.5 (all S_x state vectors)
expanding and equating they find that |a|^2 = |b|^2 = |c|^2 = |d|^2 = 0.5
This is the paragraph that confuses me--I think it might be my unfamiliarity with complex operations:
Because each coefficient is complex, each has an amplitude and phase. However, the overall phase of a quantum state vector is not physically meaningful. Only the relative phase between different components of the state vector is physically measurable. Hence, we are free to choose one coefficient of each vector to be real and positive without any loss of generality. This allows us to write the desired states as:
|+>_x = (1/√2) [ |+> + e^(iα) |->]
|->_x = (1/√2) [ |+> + e^(iβ) |->]
I am a little confused by the reasoning here. I suppose |a|^2 = (1/√2) => a = (1/√2) * e^(iθ); the absolute phase of the state vector doesn't matter, so we are free to take θ= 0 for one coefficient of each vector?
Please correct any terminology/reasoning errors I have made here.