Editing Hamiltonian (quantum mechanics) (section)

==Schrödinger Hamiltonian==

===One particle===

By analogy with [[Hamiltonian mechanics|classical mechanics]], the Hamiltonian is commonly expressed as the sum of [[Hermitian operators|operators]] corresponding to the [[kinetic energy|kinetic]] and [[potential energy|potential]] energies of a system in the form

<math display="block"> \hat{H} = \hat{T} + \hat{V}, </math>

where
<math display="block"> \hat{V} = V = V(\mathbf{r},t) ,</math>
is the [[potential energy]] operator and
<math display="block">\hat{T} = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m}\nabla^2,</math>
is the [[kinetic energy]] operator in which <math>m</math> is the [[mass]] of the particle, the dot denotes the [[dot product]] of vectors, and
<math display="block"> \hat{p} = -i\hbar\nabla ,</math>
is the [[momentum operator]] where a <math>\nabla</math> is the [[del]] [[operator (mathematics)|operator]]. The [[dot product]] of <math>\nabla</math> with itself is the [[Laplacian]] <math>\nabla^2</math>. In three dimensions using [[Cartesian coordinates]] the Laplace operator is
<math display="block">\nabla^2 = \frac{\partial^2}{ {\partial x}^2} + \frac{\partial^2}{ {\partial y}^2} + \frac{\partial^2}{ {\partial z}^2}</math>

Although this is not the technical definition of the [[Hamiltonian mechanics|Hamiltonian in classical mechanics]], it is the form it most commonly takes. Combining these yields the form used in the [[Schrödinger equation]]:

<math display="block">\begin{align}
\hat{H} & = \hat{T} + \hat{V} \\[6pt]
 & = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m}+ V(\mathbf{r},t) \\[6pt]
 & = -\frac{\hbar^2}{2m}\nabla^2+ V(\mathbf{r},t)
\end{align}</math>

which allows one to apply the Hamiltonian to systems described by a [[wave function]] <math>\Psi(\mathbf{r}, t)</math>. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

==== Expectation value ====
It can be shown that the expectation value of the Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system.

Consider computing the expectation value of kinetic energy:

<math display="block">\begin{align}
T &= -\frac{\hbar^2}{2m} \int_{-\infty}^{+\infty} \psi^* \frac{d^2\psi}{dx^2} \, dx \\[1ex]
&=-\frac{\hbar^2}{2m} \left( {\left[ \psi'(x) \psi^*(x) \right]}_{-\infty}^{+\infty} - \int_{-\infty}^{+\infty} \frac{d\psi}{dx} \frac{d\psi^*}{dx} \, dx \right) \\[1ex]
&= \frac{\hbar^2}{2m} \int_{-\infty}^{+\infty} \left|\frac{d\psi}{dx}  \right|^2 \, dx  \geq 0
\end{align}</math>

Hence the expectation value of kinetic energy is always non-negative. This result can be used to calculate the expectation value of the total energy which is given for a normalized wavefunction as:

<math display="block">E = T + \langle V(x) \rangle = T + \int_{-\infty}^{+\infty} V(x) |\psi(x)|^2 \, dx \geq  V_{\text{min}}(x) \int_{-\infty}^{+\infty} |\psi(x)|^2 \, dx  \geq V_{\text{min}}(x)  </math>

which complete the proof. Similarly, the condition can be generalized to any higher dimensions using [[divergence theorem]].

===Many particles===

The formalism can be extended to <math>N</math> particles:

<math display="block"> \hat{H} = \sum_{n=1}^N \hat{T}_n + \hat{V} </math>

where
<math display="block"> \hat{V} = V(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N,t) ,</math>
is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and
<math display="block"> \hat{T}_n = \frac{\mathbf{\hat{p}}_n\cdot\mathbf{\hat{p}}_n}{2m_n} = -\frac{\hbar^2}{2m_n}\nabla_n^2</math>
is the kinetic energy operator of particle <math>n</math>, <math>\nabla_n</math> is the gradient for particle <math>n</math>, and <math>\nabla_n^2</math> is the Laplacian for particle {{mvar|n}}:
<math display="block">\nabla_n^2 = \frac{\partial^2}{\partial x_n^2} + \frac{\partial^2}{\partial y_n^2} + \frac{\partial^2}{\partial z_n^2},</math>

Combining these yields the Schrödinger Hamiltonian for the <math>N</math>-particle case:

<math display="block">\begin{align}
\hat{H} & = \sum_{n=1}^N \hat{T}_n + \hat{V} \\[6pt]
 & = \sum_{n=1}^N \frac{\mathbf{\hat{p}}_n\cdot\mathbf{\hat{p}}_n}{2m_n}+ V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) \\[6pt]
 & = -\frac{\hbar^2}{2}\sum_{n=1}^N \frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t)
\end{align} </math>

However, complications can arise in the [[many-body problem]]. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:

<math display="block">-\frac{\hbar^2}{2M}\nabla_i\cdot\nabla_j </math>

where <math>M</math> denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as ''mass polarization terms'', and appear in the Hamiltonian of many-electron atoms (see below).

For <math>N</math> interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function <math>V</math> is ''not'' simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.

For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle,<ref name="QuantumPhysics">{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles |url=https://archive.org/details/quantumphysicsof00eisb |url-access=registration |edition=2nd |first1=R. |last1=Resnick |first2=R. |last2=Eisberg |publisher=John Wiley & Sons |year=1985 |isbn=0-471-87373-X }}</ref> that is

<math display="block"> V = \sum_{i=1}^N V(\mathbf{r}_i,t) = V(\mathbf{r}_1,t) + V(\mathbf{r}_2,t) + \cdots + V(\mathbf{r}_N,t) </math>

The general form of the Hamiltonian in this case is:

<math display="block">\begin{align}
 \hat{H} & = -\frac{\hbar^2}{2}\sum_{i=1}^N \frac{1}{m_i}\nabla_i^2 + \sum_{i=1}^N V_i \\[6pt]
 & = \sum_{i=1}^N \left(-\frac{\hbar^2}{2m_i}\nabla_i^2 + V_i \right) \\[6pt]
 & = \sum_{i=1}^N \hat{H}_i
\end{align}</math>

where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.