2.3. Postulate 3 – Measurable quantities and operators

A physically observable quantity of a quantum system is represented by a linear Hermitian operator, which implies that a measurement outcome is always a real value, not a complex number. The real values of the measurement are the eigenvalues of the Hermitian operator that describes it. The eigenvalue is the constant factor that is produced by an operation.

For a spectrum of an observable, if it's discrete, the possible results are quantized. We determine the measurable quantity by calculating the expectation value of the observable in a state as follows:

It is the sum of all the possible outcomes of a measurement of a state weighted by their probabilities. Furthermore, the state of a quantum mechanical system can be represented by the inner product of a given distance called a Hilbert space. A definition of a Hilbert space is given in Appendix A – Readying Mathematical Concepts. This definition of a state space implies the superposition principle of quantum mechanics, which is a linear combination of all real or complex basis functions :

where is the index of summation, is the total number of basis functions to obtain convergence and completeness of the wave function, and is the linear expansion coefficient, which can be real or complex numbers. Plugging the superposition principle into the definition of the expectation value, we obtain the following equation:

where we have also included the PEP. We will use the superposition principle in subsequent chapters. In this section, we present common operators and calculate the expectation value for a given system:

Section 2.3.1, Hermitian operator
Section 2.3.2, Unitary operator
Section 2.3.3, Density matrix and mixed quantum states
Section 2.3.4, Position operation with the position operators
Section 2.3.5, Momentum operation with the momentum operators
Section 2.3.6, Kinetic energy operation with the kinetic energy operators
Section 2.3.7, Potential energy operation with the potential energy operators
Section 2.3.8, Total energy operation with total energy operators

The measurable quantum quantities are derived from the classical counterparts.

2.3.1. Hermitian operator

The complex conjugate transpose of some vector or matrix often is denoted as and in quantum mechanics. The symbol is called the dagger. is called the adjoint or Hermitian conjugate of .

A linear operator is called Hermitian or self-adjoint if it is its own adjoint: .

The spectral theorem says that if is Hermitian then it must have a set of orthonormal eigenvectors:

where with real eigenvalues , and is the number of eigenvectors, and also is the dimension of the Hilbert space. Hermitian operators have a unique spectral representation in terms of the set of eigenvalues and the corresponding eigenvectors :

We revisit this topic in Section 2.3.3, Density matrix and mixed quantum states.

Writing matrices as a sum of outer products

The outer product of a ket and a bra is the rank-one operator with the rule:

The outer product of a ket and a bra is a simple matrix multiplication:

Any matrix can be written in terms of outer products. For instance, for a 2 x 2 matrix:

We will be using these matrices in Chapter 3, Quantum Circuit Model of Computation, Section 3.1.6, Pauli matrices.

2.3.2. Unitary operator

A linear operator is called unitary if its adjoint exists and satisfies , where is the identity matrix, which by definition leaves any vector it is multiplied by unchanged.

Unitary operators preserve inner products:

Hence unitary operators also preserve the norm commonly known as the length of quantum states:

For any unitary matrix , any eigenvectors and and their eigenvalues and , and , the eigenvalues and have the form and if then the eigenvectors and are orthogonal: .

It is useful to note that since for any , :

We will revisit this in Chapter 3, Quantum Circuit Model of Computation.

2.3.3. Density matrix and mixed quantum states

Any quantum state, either mixed or pure, can be described by a density matrix (), which is a normalized positive Hermitian operator where . According to the spectral theorem, there exists an orthonormal basis, defined in Section 2.3.1, Hermitian operator, such that the density is the sum of all eigenvalues ():

where ranges from 1 to , are positive or null eigenvalues (), and the sum of eigenvalues is the trace operation () of the density matrix and is equal to 1:

For example, when the density is , with , the trace of the density is:

Here are some examples of the density matrices of pure quantum states:

The density matrix of a mixed quantum state consisting of a statistical ensemble of pure quantum states , each with a classical probability of occurrence , is defined as:

where every is positive or null and their sum is equal to one:

We summarize the difference between pure states and mixed states in Figure 2.20.

Figure 2.20 – Density matrix of pure and mixed quantum states

2.3.4. Position operation

The position observable of particle has the following operators for all directions in Cartesian coordinates:

In spherical coordinates the operations become:

We can calculate the expectation value of the position for a given particle in a chosen direction with the following equation:

For example, using the same system as presented in Section 2.2.2, Probability amplitude for a hydrogen anion , the expectation value of the -position of electron 1 is determined by:

Please note that the integration over is a cubic function as opposed to a quadratic function, the integration over has an additional , and the integration over has a as compared to what is seen in the Section 2.2.2, Probability amplitude for a hydrogen anion example. In this calculation, the integration over is equal to 0, which means that the entire integration is:

This means that electron 1 is most likely to be found at the nucleus (or the origin of the coordinate system). The same holds true for , and operations.

2.3.5. Momentum operation

The component of momentum operator for particle is along the -dimension (and similarly, for the - and -dimensions) and is defined as follows in Cartesian coordinates:

We can also write these operators in terms of the spherical derivatives [ucsd]:

We can calculate the expectation value of the momentum for a given particle with the following equation:

 where we use as a generic dimension.

For example, using the same system as presented in Section 2.2.2, Probability amplitude for a hydrogen anion , the derivative for the -momentum operator of electron 1 is:

where the derivative is:

Therefore, the expectation value of the -momentum for electron 1 is:

which, due to the integration over , becomes equal to 0, as illustrated by the following SymPy code:

from sympy import symbols, sin, cos
x = symbols('x')
integrate(cos(x)*sin(x),(x,0,pi))

Here is the result:

This result is intuitive because we are in a system, which does not have momentum.

2.3.6. Kinetic energy operation

The kinetic energy operators for a single particle in a given direction in Cartesian coordinates are:

In general, kinetic energy is determined by the following in Cartesian coordinates:

and in spherical coordinates is:

We can calculate the expectation value of the kinetic energy for all the particles with the following equation:

Using the same system as presented in Section 2.2.2, Probability amplitude for a hydrogen anion , the second derivative operation for the kinetic energy of electron 1 is:

The expectation value of the kinetic energy for electron 1 is then calculated by:

where the electron mass is set to equal 1 (). The kinetic energy for electron 2 is determined with the same integrals and is equal to:

The total kinetic energy for the electrons in hydride is then the sum of the two kinetic terms:

Setting as a standard scaling, we have:

2.3.7. Potential energy operation

The potential energy, also known as Coulomb energy, relates the charge of particles and and depends on the distance between two, where . It is proportional to the inverse of the distance and is calculated as a sum over all pairs of particles in the systems:

We can calculate the expectation value of the potential energy for all the particles with the following equation:

Using the same system as presented in Section 2.2.2, Probability amplitude for a hydrogen anion , the expectation value of the potential (Coulomb) energy calculated between the two electrons is:

Now we use the Dirac delta function to approximate the inverse of :

We compute this integral with the following block of code:

from sympy import symbols, integrate, exp, DiracDelta, oo
x, y = symbols('x y')
integrate(x**2 * exp(-2*x) * integrate(y**2 * exp(-2*y)*DiracDelta(x - y),(y,0,oo)),(x,0,oo))

The result is:

Therefore, the expectation value of electron repulsion is:

The expectation value of the potential (Coulomb) energy calculated between electron 1 and the nucleus (particle 3) is:

Now we use the Dirac delta function to approximate the inverse of :

We compute this integral with the following block of code:

from sympy import symbols, integrate, exp, DiracDelta, oo
x, y = symbols('x y')
integrate(x**2 * exp(-2*x) * integrate(DiracDelta(x - y),(y,0,oo)),(x,0,oo))

Here is the result: