2.2. Postulate 2 – Probability amplitude

Consider the motion of a particle in the position space where is the position vector. The probability density of finding the particle at a particular position and at a given instant in time is calculated as a function of position: . In an orthonormal basis, the inner product of two wave functions measures their overlap. Two wave functions are orthogonal if their inner product is zero. To find the probability that a state will be found in the state upon measurement, we must compute the magnitude squared of the inner product between state and , .

The wave function in space for a multiparticle system is , with being the total number of particles, which is interpreted as the probability amplitude function at a given point in time using the following integration over the volume element for all particles in the system :

Please note we converted from Cartesian coordinates to spherical coordinates. In this setup, we can include spherical harmonic functions coupled together using CG coefficients that we discussed in the previous section in the wave function .

We will also need to include the radial wave functions. We describe how to determine the radial wave functions in Section 2.2.1, Computing the radial wave functions, and then go through an example of how to calculate the probability amplitude for a specific quantum chemistry system in Section 2.2.2, Probability amplitude for a hydrogen anion .

2.2.1. Computing the radial wave functions

The radial wave functions for hydrogen-like systems are given by:

where is calculated by:

with being the Bohr radius set equal to the Committee on Data of the International Science Council (CODATA) value in SI units, and the coefficients are defined by the following recursion relation:

for which the series terminates at . We initialize with the following Python code:

a0 = 5.29177210903e-11

The comp_ak() function has the following input parameters:

• n: Integer, principal quantum number
• l: Angular momentum quantum number with values ranging from to
• a0: Bohr radius, defined by , where is the fine structure constant, is the speed of light, and is the rest mass of the electron
• ak: Coefficient defined by the preceding recursion relation

It returns a dictionary whose keys are integers and values are the corresponding coefficients :

def comp_ak(n):
  n = max(n,1)
  # Create a dictionary with integer keys, starting with a0
  dict = {0: a0}
  for l in range (n-1):
    for k in range(n-l-1):
      ak = dict.get(k)
      #display("l: {}, k: {}, ak: {}".format(l, k, ak))
      dict[k+1] = ak*(k+l+1-n)/((k+1)*(k+2*l+2))
  return dict

Let's get the first ten coefficients:

d = comp_ak(10)
for k, v in d.items():
  print("{}, {}".format(k,v))

Here is the result:

Figure 2.19 – Coefficients that appear in the radial wave functions

Import the SymPy functions:

from sympy.physics.hydrogen import R_nl
from sympy.abc import r, Z

The sympy.physics.hydrogen.Rnl(n,l,r,Z=1) function returns the hydrogen radial wave function [SymPy_Rnl]. It has the following input parameters:

• n: Integer, principal quantum number
• l: Angular momentum quantum number with values ranging from 0 to n−1
• r: Radial coordinate
• Z: Atomic number (or nuclear charge: 1 for hydrogen, 2 for helium, and so on)

Let's try it first with :

R_nl(1, 0, r, Z)

Here's is the result:

Next with :

R_nl(2, 0, r, Z)

Here's is the result:

Last with :

R_nl(2, 1, r, Z)

Here's is the result:

2.2.2. Probability amplitude for a hydrogen anion

Let's calculate the probability amplitude at time for a hydrogen anion, also called hydride, with one proton and two electrons in a spin paired ground state.This example is for illustration purposes only, and is not meant to be a rigorous calculation.

We label the two electrons as particles 1 and 2 and choose the state where the electronic angular momentum for each electron is , and are coupled to the final or total momenta state of , where is the coupling between the angular momentum and the spin momentum. For simplicity, we assume that this system is not entangled.

We will denote the wave function with the PEP operation () as:

where the spatial function is symmetric, and the spin function is antisymmetric:

with , the CG coefficient, equal to:

Recall that we derived the antisymmetric spin state in Section 2.1.2, Fermionic spin pairing to symmetric state (), therefore we won't redo this calculation; we will simply reuse the result:

Next, we illustrate the coupling of the angular momentum spatial function for the symmetric spatial state :

with the CG coefficient equal to:

Now we plug this into the wave function:

Next, we will be using the following spherical harmonic functions:

And the radial wave function for each electron with the nuclear charge for the proton of , as determined in Section 2.2.1, Computing the radial wave functions:

The wave function for the ground state of hydride is:

The probability amplitude is calculated by determining the square of the wave function:

The integral over spin is equal to 1 due to the fact that the spin functions are normalized, resulting in:

Next, we include the PEP, where we calculate . Recall that we derived for two fermions in an antisymmetric spin state as , as shown in Figure 2.13. The operation results in a factor of :

The integral over and is equal to , illustrated with the following SymPy code:

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

Here is the result:

The integrals over and are equal to , illustrated with the following SymPy code:

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

Here is the result:

The integrals over and are equal to , illustrated with the following SymPy code:

integrate(1,(x,0,2*pi))

Here is the result:

Combining all the results, the probability amplitude is equal to 1:

Now we can move on to the rest of the postulates. Examples of these postulates will be illustrated in the following chapters of the book. As a result, we have not included code for these postulates in this chapter. We revisit this topic expectation value in Section 3.1.9, Pauli matrices.