Probability Density of Particle in Potential Well

[ronaldoshaky]

Hello I am trying to find the probability density function for a particle in a potential well of a harmonic oscillator. (My question is about complex conjugates).

I know the formula. I have to multiply $$\Psi^{*} (x, t) \Psi (x, t)$$The wave function is a linear combination of stationary states, i.e.

$$\Psi (x, t) = \frac{1}{\sqrt{2}} [ \psi_{0} (x) e^{\frac{-i \omega_{0} t}{2}} + \psi_{1} (x) e^{\frac{- 3i \omega_{0} t}{2}} ]$$

$$\psi_{0} (x)$$and $$\psi_{1} (x)$$ are real

the conjugates of $$\psi_{0} (x)$$and $$\psi_{1} (x)$$ are

$$\psi_{0}^{*} (x)$$and $$\psi_{1}^{*} (x)$$ but since the eigenfunctions are real (are the conjugates the same as the eigenfunctions), what happens when I multiply them together?

I thought that, for example,
$$\psi_{0}^{*} (x) \psi_{0} (x) = \psi_{0} (x) \psi_{0} (x)= |\psi_{0} (x) |^{2}$$

 

[SpectraCat]

Hello I am trying to find the probability density function for a particle in a potential well of a harmonic oscillator. (My question is about complex conjugates).

I know the formula. I have to multiply $$\Psi^{*} (x, t) \Psi (x, t)$$The wave function is a linear combination of stationary states, i.e.

$$\Psi (x, t) = \frac{1}{\sqrt{2}} [ \psi_{0} (x) e^{\frac{-i \omega_{0} t}{2}} + \psi_{1} (x) e^{\frac{- 3i \omega_{0} t}{2}} ]$$

$$\psi_{0} (x)$$and $$\psi_{1} (x)$$ are real

the conjugates of $$\psi_{0} (x)$$and $$\psi_{1} (x)$$ are

$$\psi_{0}^{*} (x)$$and $$\psi_{1}^{*} (x)$$ but since the eigenfunctions are real (are the conjugates the same as the eigenfunctions), what happens when I multiply them together?

I thought that, for example,
$$\psi_{0}^{*} (x) \psi_{0} (x) = \psi_{0} (x) \psi_{0} (x)= |\psi_{0} (x) |^{2}$$

Yes, that is correct, so the problem boils down to the correct handling of the time-dependent complex phases ...