An introduction to Lev Landau’s approach to Quantum Probability theory.
This introduction to Quantum Probability theory is based entirely on Landau’s Theoretical Minimum. Although this article refers to the electron in particular, it is applicable to quanta(i.e. photons, muons) in general.
Landau implicitly proposes an objective collapse theory in his description of the Measurement Problem:
The ‘classical object’ is called an apparatus, and its interaction with the environment is a measurement. By measurement, we understand any process of interaction between classical and quantum objects, occurring independently of any observer.
In spite of potential shortcomings of objective collapse theories, Landau’s analysis is based on an insightful observation of the relation between Quantum Mechanics and Classical Mechanics:
It is in principle impossible to formulate the basic concepts of Quantum Mechanics without using Classical Mechanics. As an electron has no definite path, it has no dynamical characteristics. Hence, for a Quantum System it is impossible to construct logically independent mechanics.
Quantum Mechanics can’t make completely definite predictions concerning the future behaviour of the electron.
For a given initial state of the electron, a subsequent measurement can give various results.
The essential problem in Quantum Mechanics consists in determining the probability of obtaining various Quantum Measurements.
The essence of Quantum Mechanics is that any state of a system may be described at any given moment, by a definite function \(\Psi(q)\) of the coordinates.
The square of the modulus of this function determines the probability distribution of the values of the coordinates: \(\lvert \Psi(q) \rvert^2 dq\) is the probability that a measurement performed on the system will find the values of the coordinates to be in the element \(dq\) in the configuration space.
\(\Psi\) is known as the Wave Function of the system, or a Probability Amplitude.
Let’s suppose that in a state with Wave Function \(\Psi_1(q)\) a measurement leads with certainty to a definite result, result 1, while in a state with \(\Psi_2(q)\) it leads to result 2.
Then, for any linear combination:
\[\begin{equation} c_1 \cdot \Psi_1(q) + c_2 \cdot \Psi_2(q) \tag{1} \end{equation}\]
we shall observe either result 1 or result 2.
Quantum quantities that can be measured simultaneously, but if they have definite values, no other physical quantity(not being a function of these) can have a defnite value in that state.
Let’s consider a physical quantity \(f\) which describes the state of a Quantum System. To be precise, we are generally interested not in a single specific quantity but a complete set.
The value of a physical quantity may take in Quantum Mechanics is generally known as an eigenvalue.
A complete set of these corresponds to a spectrum of eigenvalues.
In Quantum Mechanics, we may observe either a discrete or continuous spectrum of eigenvalues.
The eigenvalues of the quantity \(f\) are denoted by \(f_n\) where \(n \in \mathbb{N}\).
In particular, we may denote the Wave Function of the system in which the state where the quantity \(f\) has value \(f_n\) by \(\Psi_n\).
The Wave Functions \(\Psi_n\) are called eigenfunctions of the physical quantity \(f\).
Each of these functions are normalised so that the result of integrating over the entire configuration space yields:
\[\begin{equation} \int \lvert \Psi_n \rvert^2 = 1 \tag{2} \end{equation}\]
If the system is in an arbitrary state with Wave Function \(\Psi\), a measurement of the quantity \(f\) carried on it will give as a result one of the eigenvalues \(f_n\).
In the case of an arbitrary state, the function \(\Psi\) may be represented in the form of a series:
\[\begin{equation} \Psi = \sum_n a_n \cdot \Psi_n \tag{3} \end{equation}\]
where the \(a_n\) are constant coefficients. A set of functions in terms of which such an expansion can be made is called a complete set.
According to what has been said so far the probability is determined by an expression linear in \(\lvert \Psi \rvert^2\) and hence bilinear in \(\Psi\) and \(\Psi^*\). It follows that probabilities must be determined by expressions bilinear in \(a_n\) and \(a_n^*\).
Thus, we reach the conclusion that the squared modulus \(\lvert a_n \rvert^2\) of each coefficient in the expression (3) determines the probability of the value \(f_n\) of the quantity \(f\) in the state with wave function \(\Psi\).
As the sum of the probabilities of all possible values \(f_n\) must equal one, we have:
\[\begin{equation} \sum_n \lvert a_n \rvert^2 = 1 \tag{4} \end{equation}\]
Now, if the function \(\Psi\) were not normalised (4) would not hold and so we have:
\[\begin{equation} \sum_n a_n \cdot a_n^* = \int \Psi \Psi^* dq \tag{5} \end{equation}\]
On the other hand, multiplying by \(\Psi\) the expression \(\Psi^*=\sum_n a_n^* \Psi_n^*\) of \(\Psi^*\) and integrating we find:
\[\begin{equation} \int \Psi \Psi^* dq = \sum_n a_n^* \int \Psi_n^* \Psi dq \tag{6} \end{equation}\]
Given (5) and (6) we have:
\[\begin{equation} \sum_n a_n a_n^* = \sum_n a_n^* \int \Psi_n^* \Psi dq \tag{7} \end{equation}\]
from which we may determine the coefficients:
\[\begin{equation} a_n = \int \Psi \cdot \Psi_n^* dq \tag{8} \end{equation}\]
and if we take (3) into account we have:
\[\begin{equation} a_n = \sum_m a_m \int \Psi_m \Psi_n^* dq \tag{9} \end{equation}\]
from which we may deduce that:
\[\begin{equation} \int \Psi_m \Psi_n^* dq = \delta_{m,n} \tag{10} \end{equation}\]
where \(\delta_{m,n}=1\) when \(m=n\) and \(\delta_{m,n}=0\) otherwise.
Although \(f\) may not have a definite value, we may calculate the mean of \(f\) as follows:
\[\begin{equation} \overline{f} = \sum_n f_n \cdot \lvert a_n \rvert^2 \tag{11} \end{equation}\]
and we may define an auxiliary operator \(\hat{f}\) such that the integral of the product of \(\hat{f}\Psi\) and the complex conjugate function \(\Psi^*\) equals the mean value \(\overline{f}\):
\[\begin{equation} \overline{f} = \int \Psi^* \big(\hat{f}\Psi\big)dq \tag{12} \end{equation}\]
and using the expression (9) for \(a_n\), we may reformulate (12) as:
\[\begin{equation} \overline{f} = \sum_n f_n a_n \cdot a_n^* = \int \Psi^* \big(\sum_n a_n \cdot f_n \Psi_n \big) \tag{13} \end{equation}\]
Comparing this with (12), we find that the operator acting on \(\Psi\) has the form:
\[\begin{equation} \hat{f}\Psi = \sum_n a_n f_n \Psi_n \tag{14} \end{equation}\]
From (8), it follows that:
\[\begin{equation} \hat{f}\Psi = \int \sum_n \Psi_n^* f_n \Psi_n \Psi dq \tag{15} \end{equation}\]
so we may define the Kernel of the Operator \(K(q,q')\) as:
\[\begin{equation} \hat{f}\Psi = \int K(q,q') \Psi(q') dq' \tag{16} \end{equation}\]
\[\begin{equation} K(q,q') = \sum_n f_n \Psi_n^*(q')\Psi_n(q) \tag{17} \end{equation}\]
L. D. Landau & E.M. Lifshitz. Quantum Mechanics: Non-relativistic Theory. Pergamon Press. 2nd Edition. 1965.
Rocke (2022, March 4). Kepler Lounge: Super-determinism via Solomonoff Induction. Retrieved from keplerlounge.com
For attribution, please cite this work as
Rocke (2022, March 4). Kepler Lounge: Elements of Quantum Probability theory. Retrieved from keplerlounge.com
BibTeX citation
@misc{rocke2022elements,
author = {Rocke, Aidan},
title = {Kepler Lounge: Elements of Quantum Probability theory},
url = {keplerlounge.com},
year = {2022}
}