Undergrad Free Schroedinger equation: time separation

[chartery]

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?

 

[pines-demon]

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.

 

[chartery]

@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?

 

[div_grad]

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.