Hamiltonians

The Hamiltonian concept starts in classical physics as a way to describe the total energy of a system, and it's central to both classical and quantum mechanics.

Hamilton's Equations

In classical physics, to predict the future behavior of a system, like a swinging pendulum or a planet orbiting the sun, we use Newton's laws of motion. These laws tell us how the position and velocity of an object change over time due to forces acting on it.

However, there's another way to look at this using the Hamiltonian, which is a function that represents the total energy of the system. The Hamiltonian combines:

• Kinetic Energy (T): Energy due to motion (like how fast something is moving).
• Potential Energy (V): Energy due to position (like the height of a ball in the air).

For a simple system like a particle moving in one dimension, the Hamiltonian $H$ is:

$$H = T + V = \frac{p^2}{2m} + V(x)$$

where $p$ is momentum, $m$ is mass, and $V(x)$ is potential energy as a function of position $x$.

Using the Hamiltonian, we can derive the equations of motion (Hamilton's equations), which tell us how the position and momentum of the particle change over time.

In any of these systems, you can use the Hamiltonian to express the energy of the system. For example, if you have a pendulum, its Hamiltonian would include terms for its gravitational potential energy and kinetic energy due to its swing, described using angles and angular velocities if using polar coordinates.

Coordinate Systems

To understand any physical system in classical mechanics, we need to talk about coordinate systems:

• Cartesian Coordinates are the most common, defining positions in terms of x, y, and z directions.
• Polar Coordinates and others like cylindrical or spherical coordinates are used depending on the symmetry of the problem. These help simplify the equations of motion when dealing with circular or spherical movements.

The coordinate system forms a vector space where the Hamiltonian operates to give us the equations of motion.

When moving from classical to quantum mechanics, the Hamiltonian undergoes a transformation from a formula that computes total energy to an operator that dictates how the state of a quantum system evolves over time.

Quantization

Quantization is the process of transitioning from classical physics descriptions to quantum mechanics. This involves several key changes:

1. Variables to Operators: In quantum mechanics, physical quantities like position and momentum are no longer just numbers (as they are in classical physics); they become operators. These operators don’t just give back values; they act on quantum states, changing them according to the rules of quantum physics.
2. Poisson Brackets to Commutators: In classical mechanics, Poisson brackets are used to describe how quantities change with time. In quantum mechanics, these are replaced by commutators, which fulfill a similar role but comply with the uncertainty principle, highlighting the inherent uncertainties in measuring certain pairs of quantities simultaneously, like position and momentum.

Hamiltonian Operators

In quantum terms, the Hamiltonian’s job is to apply an energy operation to a wave function (the quantum state of a system), which describes the probability amplitudes of finding the system in different configurations. The eigenvalues (resulting from the operation of the Hamiltonian on these wave functions) represent the possible energy levels of the system, and solving these gives us a profound understanding of the system's behavior at a quantum level.

For instance, in an atom, the Hamiltonian helps calculate the energy levels accessible to an electron, which are crucial for understanding chemical properties and reactions.

Solving the Hamiltonian

In practice, solving the Hamiltonian in quantum mechanics often requires sophisticated mathematical techniques and approximations, like those mentioned earlier (Hartree-Fock, Density Functional Theory). These methods help manage the complexities of electron interactions and other quantum effects that are too complicated to solve exactly.

In summary, the Hamiltonian is a bridge between the classical and quantum worlds, providing a continuous link through its role in describing energy and dynamics, adapting its form and function to suit the needs of different physical theories and scales.

In quantum mechanics, we describe particles very differently. Instead of precise positions and velocities, we use a wavefunction $\psi$. The wavefunction gives the probability of finding a particle in a certain place at a certain time.

The Hamiltonian in quantum mechanics is an operator, not just a function. It still represents the total energy (kinetic plus potential) of the system, but it operates on the wavefunction to provide important information about the system's energy and evolution over time.

The key equation here is the Schrödinger equation: $H \psi = E \psi$ where:

• $H$ is the Hamiltonian operator.
• $\psi$ is the wavefunction.
• $E$ is the energy eigenvalue corresponding to $\psi$.

This equation tells us that applying the Hamiltonian to a wavefunction $\psi$ gives us the energy $E$ of the system times the wavefunction.

Quantum Chemistry

One practical use of this in the modern world is in quantum chemistry. Chemists use the Hamiltonian and wavefunctions to:

• Find Ground States: Determine the lowest energy state of molecules, which helps in understanding chemical reactions.
• Predict Properties: Calculate properties like bond strengths, reaction rates, and spectral lines.

For example, when we want to find the most stable structure of a molecule (its ground state), we solve the Schrödinger equation with the Hamiltonian for that molecule. This gives us the wavefunction that describes the probability distribution of the electrons in the molecule and the ground state energy.

Summary

To summarize the connection and practical use:

• Classical Physics: The Hamiltonian helps us derive equations of motion to predict how a system evolves over time.
• Quantum Physics: The Hamiltonian operator, when applied to wavefunctions, helps us understand the energy and behavior of particles on a microscopic scale.

This connection between classical and quantum descriptions through the Hamiltonian allows us to transition smoothly from understanding macroscopic objects to the strange and fascinating world of quantum mechanics, where probabilities and wavefunctions rule. And in practice, this understanding is crucial for fields like quantum chemistry, where we can predict molecular behavior and design new materials and drugs.