3.2 Conserved Properties/Constants of Motion
... only the phase changes as a function of time. A successive measurement will find always the same Eigenvalue. The
energy and the expectation value of the operator A are thus always measurable at the same time.
The state of as system is defined completely if all expectation values of those operators a ...
3.5 Why does a quantum mechanic state change?
... REMARK: Principally you do not need superconductivity to find zero ohmic resistance; once a current flow started
(described by the k-distribution of electrons, which are Eigenstates of the Hamiltonian) it would be stable for infinite
time. You need defects within the crystal or phonons to reduce the ...
... n = 2 states due to this electric field. (Give your answers in terms of e , E0 , a0 , etc.)
(b) (10) Calculate the 4 energy eigenstates as superpositions of the original unperturbed states 200 , 210 ,
211 , and 21, !1 . (I.e., if the new states are labeled 1 , 2 , 3 , 4 , you might obtain solutions ...
4.4 The Hamiltonian and its symmetry operations
... This is the adequate description of the state of a system described by the Hamiltonian H. This representation
allows to calculate the time evolution easily.
REMARK: This is just one example in natural science where discussing the symmetries serve fundamental information on the system. The search for ...
Lecture 2 Hamiltonian operators for molecules CHEM6085: Density
... • The Born-Oppenheimer approximation is based on the fact that nuclei
have much larger masses than the electrons
• To a good approximation, one can solve the Schrödinger equation only
for the electrons and assume the nuclei are frozen
• We will use this approximation from now on
• As a result, our w ...
... The similarity to the classical case is reassuring. Only off-diagonal ("nonresonant") terms can be
eliminated by a nonsingular transformation. The resulting Hamiltonian is diagonal, but
nonlinear. The generator of the transformation is determined up to a diagonal ("resonant") term.
This procedure ca ...
Homework No. 01 (Spring 2016) PHYS 530A: Quantum Mechanics II
... where v = dx/dt is the velocity in the x direction.
(a) Find the Lagrangian for this system that implies the equation of motion of Eq. (1)
using Hamilton’s principle of stationary action.
(b) Determine the canonical momentum for this system
(c) Determine the Hamilton H(p, x) for this system.
2. (10 ...
E3570: A particle on a disc with a homogeneous magnetic... levels
... E3570: A particle on a disc with a homogeneous magnetic field - Landau
Submitted by: Roee Enbar & Gil Sharon
A particle is bound to move on the XY plane in the presence of a homogeneous magnetic field
perpendicular to the plane.
(1) Write the Hamiltonian in Cartesian coordinates. ...
... A moving “particle” is described by a superposition of a great many
sin and cos waves which constructively interfere to give a Gaussian
probability near a certain point but destructively interfere everywhere else.
The Gaussian “wave packet” moves according to the kinetic energy given by
the average ...
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI-600034 M.Sc. Part-A NOVEMBER 2015
... zero point energy of H79Br.
Use the method of separation of variables to break up Schrodinger equation for a rigid
rotor into ordinary angular equations and write the solutions for each.
Show that for a 1s orbital of a hydrogen like ion, the most probable distance from the nucleus to
electron is ao/ ...
... umbrella of the correspondence principle or postulate. The latter states that a
correspondence exists between certain classical and quantum operators, (such
as the Hamiltonian operators) or algebras (such as Lie or Poisson (brackets)),
with the classical ones being in the real (R) domain, and the qu ...
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.The elementary parts of a molecule are the nuclei, characterized by their atomic numbers, Z, and the electrons, which have negative elementary charge, −e. Their interaction gives a nuclear charge of Z + q, where q = −eN, with N equal to the number of electrons. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the kinetic energies of the electrons and the Coulomb (electrostatic) interactions between the two kinds of charged particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the Coulomb Hamiltonian. From it are missing a number of small terms, most of which are due to electronic and nuclear spin.Although it is generally assumed that the solution of the time-independent Schrödinger equation associated with the Coulomb Hamiltonian will predict most properties of the molecule, including its shape (three-dimensional structure), calculations based on the full Coulomb Hamiltonian are very rare. The main reason is that its Schrödinger equation is very difficult to solve. Applications are restricted to small systems like the hydrogen molecule.Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer. The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called clamped nucleus Hamiltonian, also called electronic Hamiltonian, that acts only on functions of the electronic coordinates.Once the Schrödinger equation of the clamped nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei, an appropriate eigenvalue (usually the lowest) can be seen as a function of the nuclear coordinates, which leads to a potential energy surface. In practical calculations the surface is usually fitted in terms of some analytic functions. In the second step of the Born–Oppenheimer approximation the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on the nuclear coordinates. In the case of a breakdown of the Born–Oppenheimer approximation—which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this article for more details on this.The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory) frame, but then the translational and rotational (external) energies are not accounted for. Only the (internal) atomic vibrations enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation, which approximates the potential energy surface as a quadratic function of the atomic displacements. This gives the harmonic nuclear motion Hamiltonian. Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional harmonic oscillator Hamiltonians. The one-dimensional harmonic oscillator is one of the few systems that allows an exact solution of the Schrödinger equation.Alternatively, the nuclear motion (rovibrational) Schrödinger equation can be solved in a special frame (an Eckart frame) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for rotation, translation and vibration of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it is often referred to as Watson's nuclear motion Hamiltonian, but it is also known as the Eckart Hamiltonian.