- #1
univector
- 15
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Mathematically, I understand the following eigen equation: A|a> = a |a>, where A is an operator, |a> is the eigenstate, and a is the eigenvalue. In terms of mathematics, it is nothing more than a linear transformation.
However, physically, what does the equation mean? Is it equivalent to the following?
(1) If we measure observable A, and if the state of the system immediately before the measurement is an eigenstate |a>, then the measurement will not change the state of the system.
(2) In addition, the measurement will record a value of a for the observable A.
In general, does A |a> = |b> mean that a measurement of the observable A for a system initially in state |a> sends the system into state |b> ?
If we accept the above physical interpretation, I run into trouble in the following example. Consider the spin of an electron. Suppose initially the state is in the z+ direction: [tex]|a>=|\sigma_z+>[/tex]= (1 0)[tex]^{T}[/tex]. Note that the Pauli matrix for a measurement in the x direction is
[tex]$\sigma_x = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) = A$[/tex]
If we measurement the spin in the x direction, then the resulting state will be
[tex]$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \left( \begin{array}{c} 1 \\ 0 \end{array}\right) = \left( \begin{array}{c} 0 \\ 1 \end{array}\right) = |\sigma_z->=|b>$[/tex]
So we will observe [tex]|b>=|\sigma_z->[/tex] with certainty? But we are supposed to observe it with probability 1/2.
Where am I wrong? Thanks.
However, physically, what does the equation mean? Is it equivalent to the following?
(1) If we measure observable A, and if the state of the system immediately before the measurement is an eigenstate |a>, then the measurement will not change the state of the system.
(2) In addition, the measurement will record a value of a for the observable A.
In general, does A |a> = |b> mean that a measurement of the observable A for a system initially in state |a> sends the system into state |b> ?
If we accept the above physical interpretation, I run into trouble in the following example. Consider the spin of an electron. Suppose initially the state is in the z+ direction: [tex]|a>=|\sigma_z+>[/tex]= (1 0)[tex]^{T}[/tex]. Note that the Pauli matrix for a measurement in the x direction is
[tex]$\sigma_x = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) = A$[/tex]
If we measurement the spin in the x direction, then the resulting state will be
[tex]$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \left( \begin{array}{c} 1 \\ 0 \end{array}\right) = \left( \begin{array}{c} 0 \\ 1 \end{array}\right) = |\sigma_z->=|b>$[/tex]
So we will observe [tex]|b>=|\sigma_z->[/tex] with certainty? But we are supposed to observe it with probability 1/2.
Where am I wrong? Thanks.