1. Linear Operator
States in quantum mechanics are mathematically described as vectors in a vector space. Physical observables—the things that you can measure—are described by linear operators. Operators corresponding to physical observables must be Hermitian as well as linear.
Observables are the things you measure. For example, we can make direct measurements of the coordinates of a particle; the energy, momentum, or angular momentum of a system; or the electric field at a point in space. Observables are also associated with a vector space, but they are not state- vectors.
They are the things you measure \(\sigma_{x}\) would be an example—and they are represented by linear operators. John Wheeler liked to call such mathematical objects machines. He imagined a machine with two ports: an input port and an output port. In the input port you insert a vector, such as \(\vert A \rangle\) .
The gears turn and the machine delivers a result in the output port. This result is another vector, say \(\vert B \rangle\) .
2. Hermitian Operator
The Fundamental Theorem
- The eigenvectors of a Hermitian operator are a complete set. This means that any vector the operator can generate can be expanded as a sum of its eigenvectors.
- If and are two unequal eigenvalues of a Hermitian operator, then the corresponding eigenvectors are orthogonal.
- Even if the two eigenvalues are equal, the corresponding eigenvectors can be chosen to be orthogonal.
3. Principles
- Principle 1: The observable or measurable quantities of quantum mechanics are represented by linear operators \(\mathbf{L}\) .
We’ll soon see that \(\mathbf{L}\) must also be Hermitian. Some authors regard this as a postulate, or basic principle. We have chosen instead to derive it from the other principles. The end result is the same either way: the operators that represent observables are Hermitian.
- Principle 2: The possible results of a measurement are the eigenvalues of the operator that represents the observable. We’ll call these eigenvalues \(\lambda_{i}\) . The state for which the result of a measurement is unambiguously \(\lambda_{i}\) is the corresponding eigenvector \(\left \vert \lambda_{i} \right \rangle\) .
Here’s another way to say it: if the system is in the eigenstate \(\left \vert \lambda_{i}\right \rangle\) , the result of a measurement is guaranteed to be \(\lambda_{i}\) .
- Principle 3: Unambiguously distinguishable states are represented by orthogonal vectors.
- Principle 4: If \(\vert A\rangle\) is the state-vector of a system, and the observable \(\mathbf{L}\) is measured, the probability to observe value \(\lambda_{i}\) is
The \(\lambda_{i}\) are the eigenvalues of \(\mathbf{L}\) , and \(\left \vert \lambda_{i}\right \rangle\) are the corresponding eigenvectors.
4. Example
Now, let’s work out the details of spin operators. The first goal is to construct operators to represent the components of spin, \(\sigma_{z}, \sigma_{y}\) , and \(\sigma_{z}\) . Then we’ll build on those results to construct an operator that represents a spin component in any direction. As usual, we begin with \(\boldsymbol{\sigma}_{\boldsymbol{z}}\) .
We know that \(\sigma_{z}\) has definite, unambiguous values for the states \(\vert u\rangle\) and \(\vert d\rangle\) , and that the corresponding measurement values are \(\sigma_{z}=+\mathbf{1}\) and \(\sigma_{z}=-\mathbf{1}\) .
Here is what the first three principles tell us:
- Principle 1: Each component of \(\boldsymbol{\sigma}\) is represented by a linear operator.
- Principle 2: The eigenvectors of \(\sigma_{z}\) are \(\vert u\rangle\) and \(\vert d\rangle\) . The corresponding eigenvalues are \(+1\) and \(-1\) . We can express this with the abstract equations
- Principle 3: States \(\vert u\rangle\) and \(\vert d\rangle\) are orthogonal to each other. This can be expressed as
Recalling our column representations of \(\vert u\rangle\) and \(\vert d\rangle\) , we can write it in matrix form as
\[\left(\begin{array}{ll} \left(\sigma_{z}\right)_{11} & \left(\sigma_{z}\right)_{12} \\ \left(\sigma_{z}\right)_{21} & \left(\sigma_{z}\right)_{22} \end{array}\right)\left(\begin{array}{l} 1 \\ 0 \end{array}\right)=\left(\begin{array}{l} 1 \\ 0 \end{array}\right)\]and
\[\left(\begin{array}{cc} \left(\sigma_{z}\right)_{11} & \left(\sigma_{z}\right)_{12} \\ \left(\sigma_{z}\right)_{21} & \left(\sigma_{z}\right)_{22} \end{array}\right) \quad\left(\begin{array}{l} 0 \\ 1 \end{array}\right)=-\left(\begin{array}{c} 0 \\ 1 \end{array}\right)\]There is only one matrix that satisfies these equations. I leave it as an exercise to prove
\[\left(\begin{array}{ll} \left(\sigma_{z}\right)_{11} & \left(\sigma_{z}\right)_{12} \\ \left(\sigma_{z}\right)_{21} & \left(\sigma_{z}\right)_{22} \end{array}\right)=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)\]or, more concisely,
\[\sigma_{z}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)\]5. Summary
To summarize, the three operators \(\sigma_{x}, \sigma_{y}\) , and \(\sigma_{z}\) are represented by the three matrices
\[\begin{aligned} &\sigma_{z}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) \\ &\sigma_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \\ &\sigma_{y}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right) \end{aligned}\]If we use \(\sigma_{x}\) to measure state \(\vert up \rangle\), we will have half possibility to get value 0 and half possibility to get value 1. To be mentioned, the corresponding eigenvectors are \(\vert left \rangle\) and \(\vert right \rangle\). If we still use \(\vert up \rangle\) and \(\vert down \rangle\) to express the result, we can only get the flipped value.
Reference:
1. Nielsen, Michael A., and Isaac Chuang. "Quantum computation and quantum information." (2002): 558-559.
2. Asfaw, Abraham, et al. "Learn quantum computation using qiskit." Accessed: Oct 24 (2020): 2020.
3. Susskind, Leonard, and Art Friedman. Quantum mechanics: the theoretical minimum. Basic Books, 2014.