How to understand the physics meanings for system x--> ix?

[dongsh2]

For example, in Harmonic oscillator, we find E--> -E when x--> i x. How to understand physical meanings or in mathematics? How to understand the wave function in this case?

 

[Demystifier]

For example, in Harmonic oscillator, we find E--> -E when x--> i x.

I don't think it is true, unless one also makes the transformation p->ip.

 

[dongsh2]

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?

 

[bhobba]

A

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?

A mathematical quirk maybe - every mathematical idiosyncrasy doesn't necessarily have some deep meaning.

Thanks
Bill

 

[Demystifier]

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?