It’s a bit like measuring the weight of a person on an old-fashioned (non-digital) bathroom scale with 1 kg marks only: do we say this person is x kg ± 1 kg, or x kg ± 500 g? Do we take the half-width or the full-width as the margin of error? In short, it’s a matter of appreciation, and the 1/2 factor in our pair of uncertainty relations is not there because we’ve got two relations. We may just as well choose to equate Δ with the full-width of those probability distributions we get for x and p, or for E and t. However, the 1/2 factor in those relations only makes sense because we chose to equate the fundamental uncertainty (Δ) in x, p, E and t with the mathematical concept of the standard deviation (σ), or the half-width, as Feynman calls it in his wonderfully clear exposé on it in one of his Lectures on quantum mechanics (for a summary with some comments, see my blog post on it). We may want to think that 1/2 factor just echoes the 1/2 factor in the Uncertainty Principle, which we should think of as a pair of relations: σ x The question is: why did Schrödinger use ħ/2, rather than ħ, as a scaling factor? Let’s explore the question. Physicists don’t like adding apples and oranges. x terms in the argument of the wavefunction are both expressed as some dimensionless number, so they can effectively be added together.Note, for example, how the 1/ħ factor in ω = E/ħ and k = p/ħ ensures that the ω As a physical constant, it also fixes the physical dimensions.It fixes the numbers (so that’s its function as a mathematical constant).As a physical scaling constant, it usually does two things: Planck’s constant is, effectively, a physical scaling factor. The ħ/2 now appears as a scaling factor in the diffusion constant, just like ħ does in the de Broglie equations: ω = E/ħ and k = p/ħ, or in the argument of the wavefunction: θ = (E (ħ/m eff) to the other side, we can write it as m eff/(ħ/2).Think of the following: If we bring the (1/2) ![]() But for the rest it’s the same: the diffusion constant (D) in Schrödinger’s equation is equal to (1/2) Whatever word you prefer. □ That’s also what the presence of the imaginary unit ( i) in the equation tells us. The big difference between the wave equation and an ordinary diffusion equation is that the wave equation gives us two equations for the price of one: ψ is a complex-valued function, with a real and an imaginary part which, despite their name, are both equally fundamental, or essential. We already noted and explained the structural similarity with the ubiquitous diffusion equation in physics: Schrödinger’s equation, for a particle moving in free space (so we have no external force fields acting on it, so V = 0 and, therefore, the Vψ term disappears) is written as:
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