In quantum mechanics (and quantum field theory), eigenstates of the Hamiltonian are special quantum states that have a definite energy associated with them. Let's break this down step by step:

1. What is the Hamiltonian?

The Hamiltonian is a key operator in quantum mechanics that represents the total energy of the system. It is a function of the position and momentum operators (for non-relativistic systems) or more generally the fields and their conjugate momenta (for quantum field theory). The Hamiltonian governs the time evolution of a quantum system, according to the Schrödinger equation in non-relativistic quantum mechanics, or the Heisenberg equation in more advanced theories.

The Hamiltonian $H$ can be written in general as:

$$H = T + V$$

where:

• $T$ represents the kinetic energy of the system.

• $V$ represents the potential energy of the system.

In the context of quantum field theory (QFT), the Hamiltonian includes the energy due to the field's interactions and can be more complex, depending on the system.

2. What does it mean for a state to be an eigenstate of the Hamiltonian?

An eigenstate of an operator is a state that, when acted upon by that operator, results only in the state being multiplied by a constant (called the eigenvalue). In simpler terms, if you apply the Hamiltonian operator to an eigenstate, it just scales the state by a factor.

Mathematically, for a state $\psi$, if it is an eigenstate of the Hamiltonian $H$, then:

$$H \psi = E \psi$$

Here:

• $H$ is the Hamiltonian operator.

• $\psi$ is the eigenstate of the Hamiltonian.

• $E$ is the eigenvalue, which represents the energy associated with that eigenstate.

This equation tells us that the state $\psi$ has a definite energy $E$, and it does not change form when the Hamiltonian acts on it (it only gets multiplied by the energy $E$).

3. Example: The Particle in a Box

In the simplest quantum mechanical system, consider a particle confined in a box (also called the infinite potential well). The Hamiltonian for this system would consist of the particle's kinetic energy. The eigenstates of this system are the specific wavefunctions that satisfy the Schrödinger equation for that system. These wavefunctions correspond to the allowed energy levels of the particle.

For example, the eigenstates for a particle in a 1D box are sinusoidal functions of position (like $\sin(n\pi x / L)$, where $L$ is the size of the box and $n$ is an integer), and the energy eigenvalues are given by:

$$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$

where $\hbar$ is the reduced Planck constant, $m$ is the mass of the particle, and $n$ is the quantum number (positive integer).

4. Why are eigenstates important?

Eigenstates of the Hamiltonian are important because they represent the possible states of a quantum system that have definite energies. If a system is in one of these eigenstates, it means the system has a well-defined energy, and there is no uncertainty about its energy. This is particularly useful when studying systems in equilibrium or when performing energy measurements on a system.

For example:

• In quantum mechanics, when a system is in an eigenstate of the Hamiltonian, a measurement of its energy will always return the corresponding eigenvalue (energy).

• In quantum field theory, the concept of eigenstates is extended to fields. The vacuum state, which is the lowest energy state, and excited states, which correspond to particles or collective excitations of the field, are examples of such eigenstates.

5. Stationary States and Time Evolution

If a quantum system is in an eigenstate of the Hamiltonian, then it is said to be in a stationary state. This means the probability distribution of measurable quantities (like position or momentum) does not change over time. This is because the energy eigenvalue associated with the state doesn't change, and the state evolves only by a phase factor in time:

$$\psi(t) = \psi(0) e^{-i E t / \hbar}$$

where $E$ is the energy eigenvalue, $t$ is time, and $\hbar$ is the reduced Planck constant.

In contrast, if a system is in a superposition of eigenstates, then the system's energy is not definite, and the state will evolve over time in a more complex way.

In Summary:

• Eigenstates of the Hamiltonian are quantum states that have definite energy, and they are solutions to the time-independent Schrödinger equation (or its counterpart in QFT).

• The eigenvalue corresponding to the eigenstate is the energy of that state.

• In a stationary state, the system's properties do not change over time except for a phase factor, meaning that energy is a well-defined and stable quantity in these states.

ChatGPT said:

The Planck constant, denoted by $h$, is a fundamental constant in quantum mechanics that defines the scale of quantum effects. It relates the energy of a photon (or quantum of light) to its frequency. The Planck constant is one of the key constants that governs the behavior of matter and energy on the smallest scales of the universe.

Value:

The Planck constant is approximately:

$$h \approx 6.62607015 \times 10^{-34} \, \text{J} \cdot \text{s}$$

where:

• $J$ is the joule, the unit of energy.

• $\text{s}$ is the second, the unit of time.

Reduced Planck constant ($\hbar$):

There is also a related constant called the reduced Planck constant, denoted by $\hbar$, which is simply the Planck constant divided by $2\pi$:

$$\hbar = \frac{h}{2\pi} \approx 1.0545718 \times 10^{-34} \, \text{J} \cdot \text{s}$$

Significance:

1. Energy-Frequency Relation: The Planck constant is most famously featured in the relation between the energy of a photon and its frequency $\nu$:

   $$E = h \nu$$

   where:

   • $E$ is the energy of the photon.

   • $\nu$ is the frequency of the photon. This equation is central to quantum theory and explains how light behaves as both a particle and a wave.

2. Quantum of Action: The Planck constant is often thought of as a measure of the "quantum" of action, because it defines the smallest possible action in quantum mechanics. In classical mechanics, action is typically described by $\text{Action} = \text{energy} \times \text{time}$, but in quantum mechanics, this is quantized at the scale of the Planck constant.

3. Heisenberg Uncertainty Principle: The Planck constant appears in the Heisenberg uncertainty principle, which states that there is a fundamental limit to how precisely certain pairs of physical properties (such as position and momentum) can be known simultaneously:

   $$\Delta x \Delta p \geq \frac{\hbar}{2}$$

   where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $\hbar$ is the reduced Planck constant. This principle is a cornerstone of quantum mechanics.

4. Quantum of Angular Momentum: The Planck constant also appears in the quantization of angular momentum. In quantum mechanics, the angular momentum of a particle can only take discrete values, and these values are integer multiples of $\hbar$:

   $$L = n \hbar$$

   where $n$ is an integer.

In short, the Planck constant is crucial to understanding the quantization of energy, momentum, and other physical quantities in the quantum world. It marks the boundary between classical physics, where variables like energy and momentum are continuous, and quantum physics, where these quantities are quantized.

In Quantum Field Theory (QFT), quantum states are not necessarily stationary, but they can be stationary under certain conditions. Let's break this down:

Stationary States in Quantum Mechanics:

In traditional quantum mechanics, stationary states refer to quantum states that do not change over time except for a phase factor. These states are typically the eigenstates of the Hamiltonian, meaning they have a definite energy. When the system is in a stationary state, the probability distribution of physical observables does not change over time.

Quantum Field Theory and the Nature of States:

In QFT, quantum fields are treated as operators that act on a vacuum state (the ground state of the field), and particles are interpreted as excitations of these fields. The states in QFT can indeed be stationary in a similar sense as in standard quantum mechanics when they correspond to energy eigenstates. For example:

• A vacuum state (the state with no particles) is stationary if no external interactions are applied to it. This means the field is in its lowest energy configuration, and nothing in the field is changing over time.

• A one-particle state or a multi-particle state can also be stationary if it corresponds to a specific, time-independent energy configuration.

However, quantum states in QFT are typically not stationary in the sense that the fields evolve over time according to the quantum field equations, and interactions (like those described by the interaction Hamiltonian) lead to non-stationary behavior.

For instance, in interacting quantum field theories, like Quantum Electrodynamics (QED) or Quantum Chromodynamics (QCD), the fields and the particles associated with them are in constant interaction, and their states are evolving dynamically. These interactions mean the system does not remain in a stationary state, and the quantum fields can create and annihilate particles over time.

Time Evolution in QFT:

• The time evolution of quantum states in QFT is governed by the Hamiltonian of the system, which describes both free and interacting fields. In the case of free fields, the field behaves in a stationary manner, but as soon as interactions are introduced, the field’s configuration changes over time, leading to non-stationary states.

• A stationary state in QFT would be one where the fields are in a configuration where the probability distributions of physical observables are not changing, but this situation is typically a limiting case.

In Summary:

• Quantum states in QFT can be stationary if the system is in a state with no interactions or if it corresponds to a specific energy eigenstate.

• In most practical situations in QFT, particularly with interacting fields, quantum states are non-stationary because the fields evolve with time, leading to the creation and annihilation of particles and changes in the system’s configuration.

So, while stationary states are possible, they are not the general case in QFT, especially when dealing with real-world interacting quantum fields.

However, **quantum states in QFT are typically not stationary** in the sense that the fields evolve over time according to the **quantum field equations**, and interactions (like those described by the **interaction Hamiltonian**) lead to non-stationary behavior.

You said:

But doesn't quantization mean that all states are discontinuous?

ChatGPT said:

You're right to bring up the connection between quantization and discontinuity. Let's break this down a bit more clearly:

1. Quantization and Discontinuity:

In quantum mechanics (and more generally in quantum field theory, QFT), quantization refers to the idea that certain physical quantities, like energy, momentum, or angular momentum, can only take discrete values, rather than continuous ones. This is in contrast to classical physics, where these quantities can vary smoothly.

However, quantization doesn't necessarily mean that all properties or behaviors in quantum mechanics are discontinuous in the way you might think. The quantization often applies to specific variables (like energy levels of an electron in an atom), but it doesn't imply that everything in the system is discontinuous.

For example:

• In the energy levels of atoms, the energy can only take discrete values (quantized energy levels), but the time evolution of the state (like the wavefunction) can still be continuous. The system's state doesn't "jump" randomly or in a discontinuous way.

• For fields in quantum field theory (QFT), the fields themselves are treated as continuous, but they can be described in terms of discrete quantized excitations (which are often interpreted as particles). These excitations can interact with one another and change over time, but the field remains a continuous function.

2. Stationary vs. Non-Stationary States:

In quantum mechanics and QFT, stationary states are those where the probability distributions associated with observables do not change over time. These are typically eigenstates of the Hamiltonian (the operator that governs energy). In these states, the system remains in a fixed energy level, and properties like energy are well-defined and unchanging.

However, not all quantum states are stationary:

• Non-stationary states can exist, especially when the system is in a superposition of different energy levels or eigenstates of the Hamiltonian. These states can evolve over time, and their properties (like energy) can fluctuate because of interactions with other systems or external fields.

• In quantum field theory, fields evolve over time according to the quantum field equations. The quantum states of the field may be time-dependent because of interactions, leading to non-stationary behavior.

In short, quantization refers to the discrete nature of certain quantities (like energy or momentum), but it doesn't imply that everything about a quantum system is discontinuous. You can have continuous field dynamics that evolve in time, even if certain quantities are quantized. In fact, the field itself in QFT is continuous, while the excitations (the particles) are quantized.

So, to clarify:

• Quantization applies to certain properties (like energy levels), but the field itself (or the wavefunction in quantum mechanics) is continuous.

• Stationary states have time-independent properties, while non-stationary states evolve over time, often due to interactions, even if the system is quantized.