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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.
Schrödinger’s approach requires three quantum numbers (\(n\), \(l\), and \(m_l\)) to specify a wavefunction for the electron. The quantum numbers provide information about the spatial distribution of an electron. Although \(n\) can be any positive integer (NOT zero), only certain values of \(l\) and \(m_l\) are allowed for a given value of \(n).
Erwin Schrödinger and Werner Heisenberg devise a quantum theory. In the 1920s, physicists were trying to apply Planck's concept of energy quanta to the atom and its constituents.
We have seen that both the Schrödinger and the Heisenberg equation follows from Von Neumann’s Hilbert space formalism of quantum mechanics. Consequently, we have proved that this formalism properly unifies both Schrödingers wave mechanics, and Heisenberg, Born, and Jordans matrix mechanics.
Estimate the ground-state energy of a hydrogen atom using Heisenberg’s uncertainty principle. (Hint: According to early experiments, the size of a hydrogen atom is approximately 0.1 nm.) Strategy. An electron bound to a hydrogen atom can be modeled by a particle bound to a one-dimensional box of length \(L = 0.1 \, nm\).
The Schrödinger equation describes how the state of a system evolves. Since via experiments we have access to observables and their outcomes, it is interesting to find a differential equation that directly gives the evolution of expectation values. Heisenberg Equation.
In physics, the Heisenberg picture or Heisenberg representation [1] is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.
