Question

I can't understand this argument. I agree that the contribution due to ψ(eiS[ψ]/ℏ) must continuously go over into that due to ϕ. However, there could be a continuous function f with f(0)=0 and f(1)=π

s.t.K∼eiS[ϕ]/ℏ+⋯+eif(α)eiS[ϕα]/ℏ+⋯+eif(1)eiS[ψ]/ℏ+⋯where ϕα=0=ϕ
and ϕα=1=ψ. Thus, I think it is totally fine there is a relative phase between the contribution of ψ and ϕ although these two paths are homotopic. Why is my argument incorrect?

Answer 1

The argument suggesting a relative phase between ψ and ϕ, based on the inclusion of a continuous function f, is incorrect in the context of quantum mechanics.

Step-by-step explanation:

In quantum mechanics, the wave function can be represented as a superposition of different states, each associated with a specific phase factor. In the given argument, the student is suggesting that there can be a relative phase between the contributions of ψ and ϕ, even if they are homotopic paths. However, in quantum mechanics, the phase factors are crucial as they determine the interference patterns and probabilities of finding a particle in different states.

The argument proposes the inclusion of a continuous function f with f(0)=0 and f(1)=π to introduce a phase shift in the wave function. While this may seem reasonable mathematically, it is not physically valid. The phase factors in quantum mechanics are related to the physical properties of the system and cannot be arbitrarily introduced.

Therefore, the argument suggesting a relative phase between ψ and ϕ, based on the inclusion of a continuous function f, is incorrect in the context of quantum mechanics.