Question

It's a very basic question, where does the relation Pψ(x)=−iℏ∂xψ(x) for any square integrable ψ(x) come into existence? Some texts I found states that the above relation comes as a consequence of momentum being defined as generator of translation. But what is the basis of this definition? If momentum were defined to be generator of other form of symmetry, then it wouldn't have had the form as it does now.

In some other text, it's the other way around. Namely the action of momentum on a wavefunction is defined to be Pψ(x)=−iℏ∂xψ(x) and thence it leads to momentum being the generator of translation.

Which one is the correct one? How was such action of momentum on wavefunction historically developed?

Answer 1

The momentum operator in quantum mechanics, Pψ(x) = -iΩ∂xψ(x), comes from defining momentum as the generator of spatial translations, a concept grounded in the symmetry principles of physics. Its historical development and importance in quantum theory are linked with Noether's theorem and the probabilistic interpretation of the wave function.

Step-by-step explanation:

The relationship Pψ(x) = -iΩ∂xψ(x) is fundamental to quantum mechanics and originates from the concept of momentum as the generator of translations. This definition finds its theoretical basis in the mathematical framework of quantum mechanics, specifically through the application of operator theory to physical observables. In historical development, Noether's theorem played a crucial role by linking symmetries with conserved quantities, suggesting that the momentum operator corresponds to translational symmetry. The momentum operator, when acting on a wave function, is essential for calculating expectation values and contributes to the broader understanding of quantum behavior as outlined by the Schrödinger equation.

The behavior of particles in quantum mechanics, unlike classical mechanics, cannot be described by simple deterministic equations of motion. Instead, the wave function, denoted by ψ(x), provides probability amplitudes whose square magnitudes give the probability densities for particle positions upon measurement, as per the Born interpretation and the Copenhagen interpretation. Solving Schrödinger's time-dependent equation, which includes the momentum term, allows physicists to predict the likely behavior of a quantum system under various conditions.

Ultimately, the precise mathematical form of the momentum operator ensures that quantum mechanics aligns with the observed properties of particles at the quantum scale, including the quantization of energy levels and the probabilistic nature of quantum states, making it a cornerstone in the theory of quantum mechanics.