Causal Interpretation of Transitions in a Two-Level System

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The orthodox interpretation of transitions between quantum states in terms of discontinuous jumps is treated as an adjunct to the Schrödinger equation itself. In the causal interpretation, the particle position and momentum are well-defined and the transition can be described as a continuous evolution of the quantum particle according to the time-dependent Schrödinger equation. There are no "quantum jumps". To study transitions in a two-level system, time-dependent perturbation theory must be used. These solutions are not exact solutions of the Schrödinger equation, but they are extremely accurate. For the particular case of a two-level system perturbed by a periodic external field (but without quantization of the transition-inducing field and ignoring radiation effects), an accurate solution can be derived.


This Demonstration studies a transition from the ground state to the excited state. The quantum wave oscillating with the period between the two states, where and are time-dependent functions and is related to the strength of the external field. For the wavefunction in the ground state gives an exact solution to the Schrödinger equation. The model used is that of (spinless) quantum particles in a one-dimensional harmonic oscillator potential (described by normalized solutions of the corresponding Schrödinger equation) involving products of Hermite polynomials with a Gaussian function. According to the orthodox interpretation of quantum theory, the energy of the system, which is in a superposition of states, will not have a definite value (except for integer multiples of the period) for an ensemble of noninteracting quantum particles until a measurement is carried out. Therefore, a measurement of the energy at time will give either or , with probabilities and , respectively.

In the causal interpretation, ensembles of particles are characterized by a wavefunction, an initial position, and a trajectory. The trajectories are streamlines in the Madelung fluid, regarded as paths of quantum particles that are not directly measurable because of the perturbation caused by the measurement process. In the superposition of states, the energy for each individual quantum particle evolves in a continuous manner. The particle's motion evolves such that its energy equals or , depending on its initial position.


Contributed by: Klaus von Bloh (March 2011)
Open content licensed under CC BY-NC-SA



A two-level system is represented by a superposition of states in Dirac's bra-ket notation:


This leads to

for the ground-state wavefunction. The excited state, with energy in dimensionless units, is correspondingly


Inside the wave, the spatial starting points for the particles are linearly distributed around the peak. The velocity and quantum potential are appropriately scaled to fit.

The graphic on the right shows the squared wavefunction and the trajectories. The one on the left shows the particle positions, the squared wavefunction (blue), the velocity (green), and the quantum potential (red).

Reference: C. Dewdney and M. M. Lam, "What Happens During a Quantum Transition?," Information Dynamics (H. Atmanspacher and H. Scheingraber, eds.), New York: Plenum Press, 1991.

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