For example, David Bohm's 1952 Pilot-Wave theory provided "hidden variables" in the form of a "quantum potential" that changes instantaneously (infinitely faster than light speed) throughout all space, in order to restore a deterministic view of quantum mechanics, which Bohm thought that Einstein wanted. Einstein was appalled. Many writers describe this as the "pilot-wave interpretation" and it is both a formulation and an interpretation.

Different formulations in both classical and quantum mechanics provide exactly the same predictions and experimental results. But the mathematics of one formulation may yield solutions of a specific physical situation much more quickly and easily than others. Some are more easily generalized to relativistic mechanics. Some help us to "visualize" what is going on - the so-called "elements of reality" - in particular problems. Some are better at eigenvalues than transition probabilities, some better for bound/periodic systems than for "free" particles. Some find "constants of the motion" better. Some are better for quantum field theory than for quantum electrodynamics and so on.

Just as there are many formulations of classical mechanics, there are many formulations (some analogous) of quantum mechanics.

Bohr quantized the electron orbits (visualized as planetary electrons traveling around a Rutherford nucleus), treating them as periodic bound systems. He did not quantize the "free" radiation emitted or absorbed when electrons "jump" from one orbit to another. Bohr thought it was classical electromagnetic radiation until at least 1925, ignoring Einstein's hypothesis of light quanta (1905), their connection to waves (1909), and their emission and absorption, along with his discovery of "stimulated" emission (1916). Bohr's "correspondence principle" allowed him to match up the transitions for states with large quantum numbers to classical behavior at large distances from the nucleus, which helped him to determine some physical constants (e.g., Rydberg).

In the mid-1920's, Bohr's assistant Hendrik Kramers analyzed the allowed orbits into their Fourier components and developed a matrix of transitions between states. He showed that transition probabilites were proportional to the squares of the Fourier component waves. Werner Heisenberg helped Kramers with the calculations in a joint paper, and then returned to Göttingen where he extended the matrix idea to reformulate old quantum theory as "matrix mechanics." With the help of Max Born and Pascual Jordan, he could correctly calculate transition probabilities, explaining the strengths of spectral lines. Wolfgang Pauli used matrix mechanics to calculate the structure of the hydrogen atom, reproducing Bohr's results.

Heisenberg looked for quantitities that he called "observables," as opposed to Bohr's visualization of circular and elliptical orbits for the electrons. One of these is the system's energy, for example, which he could calculate without reference to an orbit.

Heisenberg could also calculate position and momentum "observables," but these involved non-commuting operators (for which pq ≠ qp). Max Born knew that matrices have this non-commuting property. Heisenberg showed that in general, the "quantum conditions" are that pq - qp = ih, which later was written as a minimal condition known as the "uncertainty principle," pq - qp ≥ ih.

Heisenberg's matrices are Hermitian, so its eigenvalues are real, and Heisenberg identified those eigenvalues with the possible values of an observable. The matrix elements of the Hamiltonian are diagonal (off-diagonal elements are all zero).

Identifying the energy levels E_n in an atom as observables which provide no knowledge of the internal dynamics (e.g., electron position, momentum, periodicity, etc.), Heisenberg declared the "underlying reality" as unknowable in principle. In any case, matrix mechanics gives us no "visualization" of what is going on "really."

The matrix elements of the polarization are periodic functions of the time that provide the frequencies and intensities of the spectral lines.

Heisenberg's square (n x n) matrices are operators that operate on an n x 1 single-row or -column vector. In the Heisenberg picture these "state vectors" ( | ψ > in Dirac notation) are constants, and the operators A evolve in time.

The operator H is the system Hamiltonian, the total (kinetic plus potential) energy.

For a single particle of mass m moving in an electric (but not magnetic field), Schrödinger could write his equation in ordinary physical space as

He could even write a two-particle wave function (important for the "entangled" particles in the EPR experiments), now in a six-dimensional space, δψ (r₁, r₂, t).

But Schrödinger's easily "visualizable" wave functions were not the vectors of n-dimensional "configuration space" of Heisenberg's matrix mechanics. Von Neumann called it a Hilbert space, in which n can be infinite and range over both discrete and continuous eigenvalues.

The Copenhagen-Göttingen school was interested in energy levels and "quantum jumps." Heisenberg had successfully explained the relative intensities of spectral lines in terms of transition probabilities. Pauli used matrix mechanics to derive the structure of the hydrogen atom.

Schrödinger, on the other hand, built his "wave mechanics" with an emphasis on the "wave-particle duality" that Einstein had been advocating for twenty years. For Einstein. "reality" consisted of individual light quanta emitted and absorbed by the atoms. When there are large numbers of such quanta, the wavelike properties show up. Einstein imagined the waves to be a "ghost-field" that guided the light quanta to exhibit classical interference phenomena. Crests in the waves would have more quanta than the nodes.

Louis de Broglie accepted Einstein's view that light waves consist of particles. De Broglie hypothesized that material particles might have wave characteristics that guide the particles. He proposed "pilot waves" with a wavelength related to the moomentum p of the particle.

Schrödinger's breakthrough was finding the wave equation to describe de Broglie matter waves. Schrödinger and de Broglie argued that the electronic structure of atoms could be visualized as standing waves that fit an integer number of de Broglie wavelengths around each orbit. Schrödinger claimed he had found a natural explanation for integer quantum numbers that had merely been postulated by Bohr. Einstein hoped the wave theory might restore a continuous field explanation.

Schrödinger interpreted an electron wave ψ as the actual electric charge density spread out in space. Einstein had strong reasons for objecting to this view as early as 1905 for light quanta, and Schrödinger gave up that interpretation. A few weeks after Schrödinger's final paper, Max Born offered his statistical interpretation, in which ψ is a probability amplitude (generally a complex number, which supports interference with itself), whose absolute square ψ*ψ is the probability of finding the electron somewhere. (cf. Kramers' transition amplitude, which was squared to provide the transition probability. ) Born acknowledged Einstein's similar view for the relation between light waves (the "ghost-field") and light particles (by then being called photons).

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