Solving Schrodinger’s equation gives the particle’s wave function. How are physical observables obtained from the wave functions? Use momentum, position and kinetic energy as examples to demonstrate that. The third postulate in Quantum Mechanics states that every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function. This means that the use of an operator in the equations can obtain the observables such as position, momentum, and kinetic energy from the wave functions.
The position of the particle, as well as energy, is an observable quantity which can be determined from the solutions of the time independent Schrodinger equation: HФ = EФ where H denotes the Hamiltonian operator H = – [(h2 / 2m)(d2 / dx2)] + U(x) An eigenvalue equation of the form (Operator)(Function) = (Number)(Function) applies to the TISE. The allowed values of the energy are the eigenvalues of the Hamiltonian operator, and the corresponding wave functions are its eigenfunctions.
The observable quantity (energy) is represented by an operator (the Hamiltonian). The allowed values of the observable are the eigenvalues of the operator, each corresponding to a function (the eigenfunction) which represents the state of the system when the observable has that value. Similarly, position and momentum is acted in this way, such that we could have: Observable Operator Position Momentum Energy Explain qualitatively how a scanning tunneling microscope works. The STM works by scanning a very sharp metal wire tip over a surface.
By bringing the tip very close to the surface, and by applying an electrical voltage to the tip or sample, the image can be seen in an extremely small scale down to resolving individual atoms. The STM is based on several principles: one is the quantum mechanical effect of tunneling which allows us to see the surface; another is the piezoelectric effect that allows us to scan precisely the tip with angstrom-level control; and last is a feedback loop which monitors the tunneling current and coordinates the current and the positioning of the tip.
Explain why neutral atoms in the Stern-Gerlach experiment experienced forces as they pass through the inhomogeneous magnetic field. The spin (or intrinsic angular momentum) of the atom is associated with a magnetic moment which is proportional to the spin. In an inhomogeneous magnetic field applied upon a magnetic moment, a force is aligned with the direction of the field gradient, the value of which is proportional to the field gradient and to the component of the magnetic moment in the direction of this gradient.
Thus, if the field gradient is vertical and the initial direction of the beam is horizontal, the atoms will be deflected upwards or downwards, according to the value of the component of their spin in the vertical direction. Why do they experience different forces and were thus separated into different groups according to their m values? To be specific, the atoms whose vertical spin component is positive are deflected upwards and those whose vertical spin component is negative are deflected downwards.
This is due to the field gradient and the orientation to which the beam has been applied. The angular momentum of a certain atom at ground state is characterized by the quantum number j=7/2. How many lines would you expect to see if you use these atoms in a Stern-Gerlach experiment? The nucleus of the cesium atom has spin quantum number 7/2. The total angular momentum of the lowest energy states of the cesium atom is obtained by combining the spin angular momentum of the nucleus with that of the single valence electron in the atom.
The spin produces a set of small effects in the spectra (small because of the relatively small magnetic moment), known as hyperfine structure. When nuclear spin is taken into account, the total angular momentum of the atom is characterized by a quantum number denoted by F, which for cesium, is 4 or 3. These values come from the spin value 7/2 for the nucleus and 1/2 for the electron. If the nucleus and the electron are visualized as tiny spinning tops, the value F = 4 (7/2 + 1/2) corresponds to the tops spinning in the same sense, and F = 3 (7/2 ? 1/2) corresponds to spins in opposite senses.
University/College: University of Chicago
Type of paper: Thesis/Dissertation Chapter
Date: 13 November 2016
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