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to explain a wide range of phenomena involving electrons, atoms, and light. After a great deal of effort, a new theory (together with a new law of motion) emerged in 1924. That theory is known as quantum mechanics, and it is now the basic framework for understanding atomic, nuclear, and subnuclear physics, as well as condensed-matter
In 1 dimension (2, if you count time), the equation of motion of a mass with kinetic energy K, under the influence of a time-independent potential, V (x), is, in classical physics, given by the energy balance equation: E = = K + V (x) m ̇x2 + V (x) where E, (5.1) (5.2)
SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mathematical formulation of the dynamics of a quantum system is not unique. So far we have described the dynamics by propagating the wavefunction, which encodes probability densities. This is known as the Schrödinger representation of quantum mechanics. Ultimately,
5.1 The Schr¨odinger and Heisenberg pictures Until now we described the dynamics of quantum mechanics by looking at the time evolution of the state vectors. This approach to quantum dynamics is called the Schrodinger picture.
The Heisenberg representation uses time dependent operators and constant in time states. We define the Heisenberg operator by OH(t) = U†(t)OSU(t) The two representations are clearly completely equivalent, and it is a matter of conve-nience which one is used in a given problem.
Then we discuss the evolution of state vectors and the Schr ̈odinger equation, the evolution of observables in the Heisenberg picture and the Heisenberg equations of motion, the evolution of the density operator, and the initial value problem.
(1) Schrödinger Picture: Everything we have done so far. Operators are stationary. Eigenvectors evolve under Ut(,t0). (2) Heisenberg Picture: Use unitary property of U to transform operators so they evolve in time. The wavefunction is stationary. This is a physically appealing picture, because
