Hamiltonian Operator

Definition

In quantum mechanics, the Hamiltonian operator $H$ is defined as follows.

$$ H = \dfrac{P^{2}}{2m} + V $$

Here, $P$ is the momentum operator, $m$ is the mass of the particle, and $V$ is the potential.

Explanation

It is commonly called simply the Hamiltonian. The Hamiltonian is the energy operator corresponding to the total energy of a particle in classical mechanics, that is, the sum of the kinetic energy and the potential energy, $E = \frac{p^{2}}{2m} + V$. Substituting the momentum operator $P = -\i\hbar\frac{\partial}{\partial x}$ into the definition, it is specifically as follows.

$$ H = -\dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}}{\partial x^{2}} + V $$

In three dimensions, it is as follows.

$$ H = -\dfrac{\hbar^{2}}{2m}\nabla^{2} + V $$

The eigenvalue equation of the Hamiltonian is precisely the time-independent Schrödinger equation.

$$ H \psi = E \psi $$

If the wave function $\psi$ is an eigenfunction of $H$, then the eigenvalue $E$ is interpreted as the energy of that state. The time-dependent Schrödinger equation is also expressed simply in terms of the Hamiltonian.

$$ \i\hbar \dfrac{\partial \psi}{\partial t} = H \psi $$

Since energy is an observable physical quantity, the Hamiltonian is a Hermitian operator, and therefore its eigenvalue, the energy, is always real.