The discussion centers on the transformation of variables in the context of the harmonic oscillator, specifically the implications of substituting \( x \) with \( ix \) and the resulting effects on energy and wave functions. Participants explore both the mathematical and physical interpretations of these transformations.
Participants express differing views on the necessity and implications of the transformations, particularly regarding the relationship between \( x \) and \( p \). The discussion remains unresolved, with multiple competing perspectives on the physical meanings and mathematical validity of the transformations.
Some limitations include the potential dependence on specific definitions and the unresolved nature of the mathematical steps involved in the transformations.
I don't think it is true, unless one also makes the transformation p->ip.dongsh2 said:For example, in Harmonic oscillator, we find E--> -E when x--> i x.
dongsh2 said:In harmonic oscillator, if we take x--> ix, the energy becones E_n= -(n+1/2), the wave function is changed by replacing "ix". We can check the cases n=0,1, etc. we find they indeed satisfy the Schrödinger equation.What is the reason why also p-->ip?
How do you calculate energy when you take x->-x? Do you perhaps take the Hamiltonian to act on the wave function? If yes, does that Hamiltonian depend also on p?dongsh2 said:In harmonic oscillator, if we take x--> ix, the energy becones E_n= -(n+1/2), the wave function is changed by replacing "ix". We can check the cases n=0,1, etc. we find they indeed satisfy the Schrödinger equation.
What is the reason why also p-->ip?