Hamilton operator

The Hamiltonian (also Hamiltonian ) is an operator in quantum mechanics that specifies (possible) measured energy values ​​and the time evolution . He is therefore the energy operator . For example, it provides the energy levels of the electron in the hydrogen atom . It is named after William Rowan Hamilton . The Hamiltonian formulation of classical mechanics goes back to him , in which the Hamilton function determines the time evolution and the energy. ${\displaystyle {\hat {H}}}$

Time development and energy

In quantum mechanics, each state of the physical system under consideration is indicated by an associated vector in Hilbert space . Its time evolution is determined by the Hamilton operator according to the Schrödinger equation : ${\displaystyle \psi }$ ${\displaystyle {\hat {H}}}$

${\displaystyle \mathrm {i} \,\hbar {\partial \over \partial t}\,\psi (t)={\hat {H}}\,\psi (t)}$

With

• the imaginary unit ${\displaystyle \mathrm {i} }$
• the reduced Planck's constant ${\displaystyle \hbar ={\frac {h}{2\pi }}.}$

The Hamilton operator is obtained in many cases from the Hamilton function of the corresponding classical system (with the generalized coordinate  x and the canonical momentum  p) by canonical quantization . For this purpose, the algebraic expression for the Hamilton function is read as a function of operators (position operator and momentum operator ) that satisfy the canonical commutation relations. ${\displaystyle {\mathcal {H}}(t,x,p)}$ ${\displaystyle {\hat {x}}}$ ${\displaystyle {\hat {p}}}$

However, this is not unique, since the function has the value , but the operator function has the value In addition is real , but is Hermitian . There are also quantum mechanical quantities such as spin that do not appear in classical physics . How they affect the development of time does not follow from analogies with classical physics, but must be deduced from the physical findings. ${\displaystyle x\,p-p\,x}$ ${\displaystyle 0}$ ${\displaystyle {\hat {x}}\,{\hat {p}}-{\hat {p}}\,{\hat {x}}}$ ${\displaystyle \mathrm {i} \hbar .}$ ${\displaystyle x\,p}$ ${\displaystyle {\hat {x}}\,{\hat {p}}}$

The eigenvalue equation

${\displaystyle {\hat {H}}\,\varphi _{E}=E\,\varphi _{E}}$

determines the eigenvectors of the Hamilton operator; they are stationary for a time-independent Hamilton operator , i.e. H. independent of time in every observable property. The eigenvalues are the associated energies. ${\displaystyle \varphi _{E}}$ ${\displaystyle E}$

Since the Hamilton operator is Hermitian (more precisely, essentially self-adjoint ), the spectral theorem states that the energies are real and that the eigenvectors form an orthonormal basis of the Hilbert space. Depending on the system, the energy spectrum can be discrete or continuous. Some systems, e.g. B. the hydrogen atom or a particle in the potential well , have a downwardly restricted , discrete spectrum and above that a continuum of possible energies.

The Hamilton operator generates the unitary time development . If for all time and between and the Hamiltonian with commutes , thus effecting ${\displaystyle \tau }$ ${\displaystyle \tau '}$ ${\displaystyle t_{0}}$ ${\displaystyle t}$ ${\displaystyle H(\tau )}$ ${\displaystyle H(\tau ')}$

${\displaystyle {\hat {U}}(t,t_{0})=\exp \left(-{\frac {\mathrm {i} }{\hbar }}\int _{t_{0}}^{t}{\hat {H}}(\tau )\,\mathrm {d} \tau \right)}$

the unitary mapping of each initial state to its associated state at the time ${\displaystyle \psi (t_{0})}$ ${\displaystyle \psi (t)=U(t,t_{0})\,\psi (t_{0})}$ ${\displaystyle t.}$

If the Hamilton operator does not depend on time ( ), this simplifies to ${\displaystyle {\hat {H}}\neq f(t)}$

${\displaystyle {\hat {U}}(t,t_{0})=\exp \left(-{\frac {\mathrm {i} }{\hbar }}{\hat {H}}(t-t_{0})\right).}$

Operators that swap with are conserved quantities of the system in the case of a time-independent Hamilton operator, especially the energy. ${\displaystyle {\hat {H}}}$

An energy-time uncertainty relation also applies to the energy , but in quantum mechanics one has to proceed differently when deriving it than, for example, in the position-momentum uncertainty relation .

Examples

Quantum mechanical particle in potential

From the Hamilton function

${\displaystyle {\mathcal {H}}\left({\mathbf {x} },{\mathbf {p} }\right)={\frac {{\mathbf {p} }^{2}}{2\,m}}+V({\mathbf {x} })}$

for a non- relativistic , classical particle of mass that moves in potential , a Hamilton operator can be read off. To do this, the expressions for the momentum and the potential are replaced by the corresponding operators: ${\displaystyle m}$ ${\displaystyle V(\mathbf {x} )}$

${\displaystyle {\hat {H}}({\hat {\mathbf {x} }},{\hat {\mathbf {p} }})={\frac {{\hat {\mathbf {p} }}^{2}}{2\,m}}+V({\hat {\mathbf {x} }}).}$

In the position representation, the momentum operator acts as a derivative and the operator as a multiplication with the function. The application of this Hamiltonian of a point particle of mass in the potential to the spatial wave function of the particle has an effect ${\displaystyle {\hat {\mathbf {p} }}}$ ${\displaystyle -\mathrm {i} \hbar {\tfrac {\partial }{\partial \mathbf {x} }}}$ ${\displaystyle V({\hat {\mathbf {x} }})}$ ${\displaystyle V(\mathbf {x} ).}$ ${\displaystyle m}$ ${\displaystyle V(\mathbf {x} )}$ ${\displaystyle \Psi }$

${\displaystyle \Rightarrow {\hat {H}}\Psi (\mathbf {x} )={\Bigl (}-{\frac {\hbar ^{2}}{2\,m}}\Delta +V(\mathbf {x} ){\Bigr )}\Psi (\mathbf {x} ).}$

Where is the Laplace operator . ${\displaystyle \Delta ={\tfrac {\partial ^{2}}{\partial x^{2}}}+{\tfrac {\partial ^{2}}{\partial y^{2}}}+{\tfrac {\partial ^{2}}{\partial z^{2}}}}$

The Schrödinger equation is thus

${\displaystyle \mathrm {i} \,\hbar \,{\frac {\partial }{\partial t}}\Psi (t,\mathbf {x} )=-{\frac {\hbar ^{2}}{2\,m}}\Delta \Psi (t,\mathbf {x} )+V(\mathbf {x} )\cdot \Psi (t,\mathbf {x} ).}$

This Schrödinger equation of a point mass in the potential is the basis for explaining the tunnel effect . It provides for insertion of the Coulomb potential (as a potential for interaction between an electron and a proton ), the spectral lines of the hydrogen - atom . By using the appropriate potentials, the spectral lines of other light atoms can also be calculated.

One-dimensional harmonic oscillator

→ Main article : Harmonic oscillator (quantum mechanics)

Analogously, one obtains the Hamilton operator for the quantum mechanical harmonic oscillator, which can only move along a line

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {1}{2}}m\,\omega ^{2}\,x^{2}.}$

The energies can be determined algebraically. You get

${\displaystyle E_{n}=E_{0}+n\,\hbar \omega ,\quad n\in \{0,1,2,\dots \}.}$

These are the same energies as that of a basic state with energy to which a quantum of energy has been added. ${\displaystyle E_{0}}$ ${\displaystyle n}$ ${\displaystyle \hbar \,\omega }$

Spin in a magnetic field

The Hamilton operator belongs to the spin of an electron that is bound to an atom and is in an unpaired state (only in the electron cloud ) in the magnetic field ${\displaystyle \mathbf {S} }$ ${\displaystyle \mathbf {B} }$

${\displaystyle {\hat {H}}=-\gamma \mathbf {S} \cdot \mathbf {B} .}$

It is

• ${\displaystyle \gamma }$the gyromagnetic ratio of the electron
• ${\displaystyle \mathbf {S} }$the spin operator .

Since the spin in the direction of the magnetic field can only assume the eigenvalues or ( spin polarization ), the possible energies are . In the inhomogeneous magnetic field of the Stern-Gerlach experiment , a particle beam of silver atoms splits into two partial beams. ${\displaystyle \hbar /2}$ ${\displaystyle -\hbar /2}$ ${\displaystyle \pm {\frac {\gamma }{2}}\,|\mathbf {B} |}$

Charged, spinless particle in an electromagnetic field

The Hamilton operator of a particle with a charge in an external electromagnetic field is obtained by minimal substitution ${\displaystyle q}$

${\displaystyle {\hat {H}}={\frac {1}{2m}}{\bigl (}{\hat {\mathbf {p} }}-q\mathbf {A} (t,{\hat {\mathbf {x} }}){\bigr )}^{2}+q\,\varphi (t,{\hat {\mathbf {x} }}).}$

Marked here

• ${\displaystyle \mathbf {A} (t,{\hat {\mathbf {x} }})}$the vector potential
• ${\displaystyle \varphi (t,{\hat {\mathbf {x} }})}$the scalar potential .

When multiplying the brackets, it should be noted that and because of the location dependence of in general do not commute. This is only the case in the Coulomb calibration . ${\displaystyle {\hat {\bf {p}}}}$ ${\displaystyle {\bf {A}}({\hat {\bf {x}}})}$ ${\displaystyle {\bf {A}}}$

See also

• Darwin term

literature

• Peter Rennert, Angelika Chassé and Wolfram Hergert: Introduction to Quantum Physics. Experimental and theoretical basics with exercises, solutions and Mathematica notebooks. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00769-0 .