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The Power of Phase Space in Dynamic Systems and Quantum Mechanics
Understanding how the concept of Phase space helps us navigate dynamic systems and the crucial role it plays in Quantum Mechanics
A crucial concept in both classical and Quantum physics, is the tracking of objects (or observables) in space. Understanding the position, trajectory, velocity, momentum, etc., all play an integral role in the understanding of dynamic systems. From tracking the path of a planet’s orbit to tracking the path of an electron, everything requires a means of measurement and a comprehensive understanding of all the features that go into tracking the object.
In this article, we shall explore how physicists developed the concept of Phase space to track objects in a dynamic system. We shall explain this concept both in terms of classical physics as well as Quantum physics while discussing the importance of its role. Let’s jump right in!
Tracking the position and trajectory of an object
Physicists track the position and trajectory of an object in space using various coordinate systems, such as Cartesian coordinates and radius vectors, depending on the specific context and requirements of the problem.
In Cartesian coordinates, a three-dimensional space is represented by three perpendicular axes: x, y, and z. To track the position of an object, physicists measure its distances along each of these axes. For example, if we consider a point in space, its position can be specified by three numbers (x, y, z) that represent the distances along the x-axis, y-axis, and z-axis, respectively. By continuously measuring these values over time, physicists can track the position and determine the object's trajectory.
Another commonly used coordinate system is the radius vector, which is particularly useful when dealing with spherical or polar coordinates. In this system, the position of an object is described using a distance (r) from a reference point, such as the origin, and angles (θ and φ) that specify the direction. The angle θ represents the inclination or elevation from a reference plane, while the angle φ denotes the rotation or azimuthal angle around a reference axis. By updating the values of r, θ, and φ as the object moves, physicists can precisely track its position and trajectory.
In the previous weeks, I wrote a detailed article on the tracking of objects using the Cartesian coordinates system. You can check that out to get a more in-depth understanding of these concepts.
The problem with tracking objects in a dynamic system
In real-time dynamic systems, tracking the position and trajectory of an object can be a messy process. And that makes sense. Let’s think about it for a second.
Tracking an object that is following a predictable trajectory at fixed velocity and momentum over time is a straight-forward process. The data across time-stamps follows an expected path with predictable increments or decrements in velocity and momentum. Now, how many real-world examples can you think of that follow such a pattern? Maybe a handful. If the majority of the examples followed such a pattern, then calculus would be a useless subject.
In the real world, most objects are a part of a dynamic system with multiple features playing a role in determining their position and trajectory. The velocity, position, trajectory, and momentum of the object may all change at regular or irregular time intervals in ways that may or may not be predictable.
The concept of tracking observables in a dynamic system takes a whole different and way more complex tone when you start considering Quantum mechanics in which we think not in terms of certainties but probabilities.
In order to tackle this complexity and come up with a more comprehensive means of tracking an object (or observable), physicists came up with a method of measurement and tracking called as Phase space.
What is Phase space?
Phase space is a concept used in physics to describe all the possible states that a system can be in. The concept of phase space in physics was developed by the French mathematician Joseph Fourier and later expanded upon by the French mathematician and physicist Pierre-Simon Laplace in the late 18th and early 19th centuries.
However, the modern understanding and formalisation of phase space emerged in the late 19th century with the work of the Austrian physicist Ludwig Boltzmann and the French mathematician Poincaré. Boltzmann's statistical mechanics and Poincaré's studies on celestial mechanics greatly contributed to the development of the concept of phase space as a mathematical tool for describing the behaviour of dynamical systems.
Let us try to understand this concept with the help of an example.
Imagine you have a system, like a swinging pendulum or a bouncing ball. In order to understand and predict how these systems behave, we need to keep track of certain properties, such as the position and momentum of the pendulum or the position and velocity of the ball.
Now, instead of thinking about just one snapshot of the system's properties, phase space expands our perspective. It considers all the possible combinations of these properties that the system can have. For example, the pendulum can be at different angles and have different momentums, while the ball can be at various positions and have different velocities.
To visualise this, we imagine a space with axes representing the different properties of the system. For the pendulum, we can have one axis representing the angle and another axis representing the momentum. For the ball, one axis represents the position and the other represents the velocity. Each point in this space corresponds to a specific combination of properties, representing a particular state of the system.
In this phase space, we can track how the system evolves over time. By observing how the points move, we gain insights into the behaviour of the system. We can see if it follows a predictable path or if it becomes chaotic and unpredictable.
Problems that Phase space solves
Phase space addresses several problems in the study of dynamical systems. Listed below are the three major problems with respect to tracking objects in space.
Understanding complex motion
Phase space helps in understanding and representing the complex motion of objects in a dynamic system. It provides a comprehensive description of the system's states, including position, velocity, and other relevant parameters. By visualising the trajectories and patterns in phase space, researchers can gain insights into the intricate motion of objects.
Associating data with tracking algorithms
Phase space does not directly address the problem of data association. It focuses on representing the states of the system rather than associating measurements with specific objects. However, phase space can be used in conjunction with tracking algorithms that handle data association to improve the accuracy and reliability of object tracking.
Real-Time analysis of states of change
Phase space can be leveraged in the development of tracking algorithms that operate in real-time. By utilising phase space representations, algorithms can efficiently process incoming data and update the object's state in a timely manner.
Phase space in Quantum Mechanics
In classical mechanics, phase space describes the set of all possible states of a system, represented by pairs of positions and momenta. However, in quantum mechanics, the concept of phase space undergoes modifications due to the inherent probabilistic nature of quantum phenomena.
In quantum mechanics, the phase space is replaced by a mathematical construct known as the "phase space distribution" or "Wigner function".
The Wigner function describes the probability density of finding a quantum system in a particular state in phase space. Unlike classical mechanics, where phase space points correspond to precise states, the Wigner function assigns probabilities to regions in phase space rather than individual points.
The Wigner function allows for the calculation of various quantities related to the system's dynamics, such as expectation values of observables and time evolution. It provides a valuable tool for analysing quantum systems and understanding their behaviour, including phenomena like quantum interference and entanglement.
The Wigner function
Think of the Wigner function as a map that assigns probabilities to different regions in phase space. It tells you how likely the particle is to be found in a specific position with a specific momentum. Instead of focusing on precise points, the Wigner function looks at larger regions in phase space and assigns probabilities to those regions.
Breaking down the Wigner function
The Wigner function can look long and super complicated. At least that’s how I felt the first time I looked at it. But when you break down the equation into simpler bits, it starts to make sense. Here's a step-by-step breakdown of its formal definition:
1. Consider a quantum system, such as a particle, described by its quantum state |ψ⟩.
2. Define the Wigner function, W(q, p), where q represents the position coordinate and p represents the momentum coordinate.
3. To calculate the Wigner function, start by choosing a basis for the Hilbert space (a vector space equipped with an inner product that satisfies certain properties) of the system. This basis consists of a set of states |n⟩, where n is an index that labels the states.
4. Express the Wigner function in terms of the density matrix ρ, which represents the quantum state |ψ⟩. The density matrix ρ is given by ρ = |ψ⟩⟨ψ|.
5. Expand the density matrix in the chosen basis, ρ = Σn,m ρnm |n⟩⟨m|, where ρnm represents the matrix elements of the density matrix.
6. The Wigner function is then given by the formula:
W(q, p) = (1/πħ) ∫ dy e^(-2ipy/ħ) Σn,m ρnm e^(2iq(n-m)ħ),
where ħ is the reduced Planck's constant.
7. Calculate the integral over the variable y, which represents the phase space variable conjugate to momentum p. This integral sums up the contributions from different quantum states and determines the probabilities in phase space.
8. The resulting expression for the Wigner function provides a distribution of probabilities in phase space, assigning values to different combinations of position q and momentum p.
And that’s it! Now you have a phase space to work with. I know that a lot of the jargon may not have made sense despite my best efforts at simplifying it. That is perfectly understandable. Quantum mechanics is a strange world and it takes some time getting used to working with it.
My recommendation is to spend time exploring the intricacies of each step I mentioned above. You can try to go more in-depth into concepts such as the Hilbert space and density matrix. Inshallah, I shall myself write more articles in the future, exploring these concepts.
Phase space is a fundamental concept in physics that allows us to understand and analyse the behaviour of dynamical systems. By representing the states of a system in a multidimensional space, phase space provides a comprehensive description of the system's properties and their evolution over time.
Through the concept of phase space, physicists can address a range of problems encountered in the study of dynamical systems. It provides a framework for dealing with uncertainties and noise in real-time tracking, modelling the nonlinear dynamics and chaotic behaviour of objects in space, and understanding the conservation laws and symmetries governing physical systems.
Phase space also aids in the analysis of phase transitions, equilibrium states, and the long-term behaviour of systems. It allows scientists to identify stability regions, periodic orbits, and chaotic regimes, enabling predictions and insights into the future behaviour of complex systems.
By employing mathematical tools and techniques within phase space, researchers can overcome challenges in tracking objects, studying celestial phenomena, and investigating the behaviour of particles and systems at various scales. Phase space offers a holistic perspective, encompassing all relevant variables and providing a visual representation of the system's dynamics.
As we continue to explore the mysteries of the universe, phase space remains an invaluable tool, facilitating the understanding of fundamental principles and enhancing our ability to predict, analyse, and make informed decisions in various fields of physics. It’s applications extend beyond classical mechanics, encompassing quantum mechanics, statistical mechanics, and other branches, where it continues to guide researchers towards deeper insights into the workings of the physical world.
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