I Free Schroedinger equation: time separation

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Hi

Apologies for formatting, I can't get PF's new Tex to work for me.

The (?most) general solution of the free Schroedinger eq. is e^{±i(kx-ωt)} , which implies e^{+iωt} should solve the time separated ODE:
dψ/dt = -iωψ, but instead it satisfies dψ/dt = +iωψ which is not obviously (to me) the same equation. Could someone explain where my logic errs please?
 
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chartery said:
Hi

Apologies for formatting, I can't get PF's new Tex to work for me.

The (?most) general solution of the free Schroedinger eq. is e^{±i(kx-ωt)} , which implies e^{+iωt} should solve the time separated ODE:
dψ/dt = -iωψ, but instead it satisfies dψ/dt = +iωψ which is not obviously (to me) the same equation. Could someone explain where my logic errs please?
Because the energy ##E=\hbar \omega## is supposed to be positive, physically speaking (up to an additive constant).
Insert ##\psi (x,t)=e^{\pm i\omega t}\psi(x)## in Schrödinger's equation and see what happens.
 
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@pines-demon, thanks for reply. If I plug in e+iωt , I get separation constant as -ħω. Does that mean it is a "negative energy" solution?
 
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chartery said:
If I plug in e+iωt , I get separation constant as -ħω. Does that mean it is a "negative energy" solution?
Note that the wave functions in the Schrödinger equation are complex functions.

##e^{-\mathrm{i}\omega t}## corresponds to the positive-energy solution.

##e^{+\mathrm{i}\omega t}=\left(e^{-\mathrm{i}\omega t}\right)^*## then corresponds to complex conjugate of the positive-energy solution.

For a free particle, both "##\pm##" solutions correspond to positive energies, its just that you use both the "normal" wave function ##\psi## and the complex-conjugated wave function ##\psi^*##.

This plays a role in quantum field theory, where ##\psi##'s correspond to wave functions of particles and ##\psi^*##'s - to wave functions of their anti-particles. Both the particle and its anti-particle are physical and have positive energies; the difference is that one of the wave functions is a complex-conjugated version.
 
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