The discussion centers on the transformation of variables in the context of the harmonic oscillator, specifically the implications of substituting x with ix. This transformation results in energy values of E_n = -(n + 1/2) and modifies the wave function accordingly. The necessity of also transforming momentum (p to ip) is highlighted, suggesting a deeper mathematical relationship. The participants confirm that these transformations satisfy the Schrödinger equation for various quantum states.
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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?