The Hamiltonian is one of the most important concepts in Quantum Mechanics.
Definition: The Hamiltonian of any system is an operator (or function) corresponding to the total energy of the system.
Every system is different, and hence can have its own functions or rules to describe the energies of its particles. Hence the Hamiltonian of different systems could be very different. The particles of the system could be in different energy states at different times. When this energy is measured - there could be different outcomes each time. The set of all possible energy outcomes is called a system's energy spectrum.
Most importantly - The Hamiltonian allows us to describe the dynamics of a system and predict the behavior of a wide range of phenomena (systems) in nature.
It is named after the physicist Sir William Rowan Hamilton, who developed the formalism in the 19th century. It is used in both classical and quantum mechanics,
In Classical Mechanics
In classical mechanics, the Hamiltonian is a function that describes sum of kinetic and potential energies of a physical system. It is typically denoted by H and is expressed in terms of the generalized coordinates and momenta of the system. That is the sum of the kinetic and potential energies. The kinetic energy term represents the energy associated with the motion of the particles, while the potential energy term accounts for the forces acting within the system.
Mathematically, the Hamiltonian can be expressed as:
H = T + V,
where T is the kinetic energy and V is the potential energy.
So, if we know both T and V - we can predict the behavior of a system.
In Quantum Mechanics
While the above equation fits well for a classical model, In the quantum realm - the operator or the rule must account for wave-particle duality and wave-like properties of matter. This is where Schrödinger's equation comes into the picture. The Hamiltonian plays a crucial role in the Schrödinger equation, which describes the time evolution of quantum systems.
Wave function
The wave function, denoted by Ψ (psi), contains information about position, momentum and other properties of a quantum system, and is used to calculate the probabilities of different outcomes. It represents the state of the system or its probability density function. In simpler terms - what are the different probabilistic outcomes which are expected for the wave output. For example, by finding the wave function of an electron in an atom, we can calculate the probabilities of finding the electron in different locations around the nucleus.
The Schrödinger's equation
In its simplest form - Schrödinger's equation, or Schrödinger's wave equation, can be understood as an equation that relates the energy of a system to its wave function.
Schrödinger's equation comes into two forms - time-independent and time-dependent. The time-independent version provides solutions that correspond to specific energy levels, while the time-dependent version of the equation describes how the wave function evolves with time.
The time-independent form of Schrödinger's equation is written as:
ĤΨ = EΨ
where Ĥ is the Hamiltonian operator (analogous to the classical Hamiltonian), Ψ is the wave function, E is the energy of the system,
If we look closely - the above equation looks like an eigen value equation. Hence when the Hamiltonian operator acts on the wave function Ψ, it yields a new wave function that is proportional to Ψ. The constant of proportionality is the energy E. In other words, the equation describes how the wave function changes in response to the energy of the system.
The Hamiltonian operator - Ĥ, and its eigenvalues correspond to the "quantized" energy levels of a quantum system. Since quantum physics is about measurement of the energy spectrum and its time evolution - the Hamiltonian becomes a very important concept.
Solving Schrödinger's equation allows us to determine the wave function Ψ and, consequently, the probabilities associated with different physical observables.
The time-dependent Schrödinger equation is represented as
or
The time-dependent Schrödinger equation in position basis is given as
Where,
i - Imaginary unit,
Ψ - Time-dependent wavefunction,
h2 - h-bar (planck's constant),
V(x) - potential
Ĥ = Hamiltonian operator.
Schrödinger's equation provides a mathematical framework for understanding the behavior of quantum systems and has been successful in predicting and explaining a wide range of phenomena at the quantum level.
Given the state at some initial time (t=0), we can solve it to obtain the state at any subsequent time. In particular, if Ĥ is independent of time, then
So this equation tells us that if we know the wave function at time t = 0, we can find the wave function at any time t using the above equation.
is a unitary operator. It is the time evolution operator of a closed quantum system.
References
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