What is the Hamilton operator for hydrogen?

Hamilton operator

The Hamilton operator$ \ has H $ determined in quantum mechanics the time evolution and the possible energy measurements of the associated physical system, for example the electron in the hydrogen atom. It is named after William Rowan Hamilton, to whom the Hamiltonian formulation of mechanics goes back, in which the Hamilton function determines the time evolution and the energy.

Time development and energy

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

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

With

The Hamilton operator is obtained in many cases by what are known as canonical quantization from the Hamilton function $ \ mathcal H (t, x, p) $ of the corresponding classical system (with the generalized coordinate x and the canonical momentum p). For this purpose, the algebraic expression for the Hamilton function is read as a function of operators (position operator $ \ hat x $ and momentum operator $ \ hat p $) that satisfy the canonical commutation relations. However, this is not unique because the function $ x \, p -p \, x $ has the value $ 0 $, the operator function $ \ has x \, \ has p - \ has p \, \ has but the Value $ \ mathrm i \ hable. $ In addition, $ x \, p $ is real, but $ \ has x \, \ has p $ not 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 has to be deduced from the physical findings.

The eigenvalue equation

$ \ has H \ varphi_E = E \ varphi_E $

determines the eigenvectors $ \ varphi_E $ of the Hamilton operator. If the Hamilton operator is independent of time, they are stationary, i. H. independent of time in every observable property. The eigenvalues ​​$ E $ are the associated energies.

Since the Hamilton operator is Hermitian (more precisely, essentially self-adjoint), the spectral theorem says 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, for example the hydrogen atom or a particle in the potential well, have a discrete spectrum that is limited downwards and above that a continuum of possible energies.

The Hamilton operator generates the unitary time development. If for all times $ \ tau $ and $ \ tau '$ between $ t_0 $ and $ t $ the Hamilton operator $ H (\ tau) $ commutes with $ H (\ tau') $, then causes

$ \ 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 $ \ psi (t_0) $ to the associated state $ \ psi (t) = U (t, t_0) \ psi (t_0) $ at time $ t. $

If the Hamilton operator does not depend on time, this simplifies to

$ \ hat U (t, t_0) = \ exp \ left (- \ frac {\ mathrm i} {\ hbar} (t - t_0) \ hat H \ right). $

Operators that swap with $ \ hat H $ are conserved quantities of the system in the case of a time-independent Hamilton operator. In particular, the energy is then a conserved quantity.

Examples

Quantum mechanical particle in potential

From the Hamilton function

$ \ 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 $ m $, which moves in the potential $ V (\ mathbf x) $, a Hamilton operator can be read off. To do this, the expressions for the momentum and the potential are replaced by the corresponding operators:

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

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

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

The Schrödinger equation is thus

$ \ 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. At the onset of the Coulomb potential (as the potential for the interaction between an electron and a proton), it supplies the spectral lines of the hydrogen atom. By using appropriate potentials, the spectral lines of other light atoms can also be calculated.

One-dimensional harmonic oscillator

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

$ \ 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

$ E_n = E_0 + n \, \ hbar \ omega, \ quad n \ in \ {0,1,2, \ dots \}. $

These are the same energies as those of a ground state with energy $ E_0 $, to which a quantum of the energy $ \ hbar \, \ omega $ was added $ n $ times.

Spin in a magnetic field

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

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

It is

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

Charged, spinless particle in an electromagnetic field

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

$ \ 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

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

When multiplying the brackets, note that the operators $ \ hat {\ mathbf {p}} $ and $ \ mathbf {A} (t, \ hat {\ mathbf {x}}) $ only swap for Coulomb calibration.

See also