In quantum mechanics, the superposition of states collapses when measured. This behaves as if nature cares only when you are looking. This is odd and we may expect the math will be bizarre also. In reality, the math is amazingly simple and elegant. A wave function |ψ⟩ collapses when measured and all measurements have an associated operator U (observable).
After we calculate the observable, we use it to make measurements. In this article, we will detail how both are done.
Eigenvalues & eigenvector
In linear algebra, λ (a scalar) and v (a vector) are the eigenvalue and eigenvector of A if
For a matrix A, it can have multiple eigenvalues and eigenvectors.
By experiment, we know the spin of a particle can be measured in two unambiguous distinguishable states|u⟩ and |d⟩ with measured value +1 and -1 respectively. These vectors are orthogonal to each other. If we are given a particle in either state, we can always set up an experiment to distinguish them without ambiguity. By the principle of quantum dynamics, these are our eigenvectors and eigenvalues for our observable σ, i.e.
Substitute |u⟩ and |d⟩ with the equations above, it becomes:
The solution for these equations are:
This is the observable along the z-axis. It is this simple! Let’s repeat the calculation for the x-axis.
By experiment, when a particle is prepared to be right spin, it has half of the chance to be measured as up spin or down spin. So we can express the right and left spin as a superposition of up and down spin below.
Again, |r⟩ and |l⟩ are unambiguous distinguishable when measured along the x-axis with measurements +1 and -1 respectively. So,
The solution is:
We repeat the calculation with the y-axis. Here are the observables along all three axes:
Once we have the observable, we can use them to make measurements.
So for an observable, what is the value when a quantum state is measured. Let’s start with X which is the observable along the x-axis.
The two eigenvalues of this observable are 1 and -1 with the corresponding eigenvectors:
Let’s rewrite the ket and the bra of e0 in the matrix form:
with the definition:
The probability of measuring |ψ⟩ with a specific eigenvalue value is:
So for state |0⟩ to be measured with eigenvalue 1, the probability is:
The state after the measurement is:
i.e. the states measured with eigenvalue 1 and -1 are:
As visualized below, if |0⟩ is measured along the x-axis, it has an equal chance to be measured as |r⟩ or |l⟩. The quantum state becomes:
For a particular state Ψ, the average value for the observable A when measured is:
Operators corresponding to physical observables are Hermitian. An operator is a Hermitian if it equals its transpose after taking a complex conjugate. Hermitian operators guarantee to have real eigenvalues. i.e. its measured values are real. The diagonal value of a Hermitian operator has to be real and the transposed elements are its complex conjugate.
Quantum mechanics is non-intuitive. But the beauty is it has a very elegant and amazing simple model based on math. Hope you have enjoyed the math here.
Credit and reference
Susskind, Leonard, Friedman, Art. Quantum Mechanics: The Theoretical Minimum.
Jurgen Van Gael: The Role of Interference and Entanglement in Quantum Computing.