One-dimensional Schrodinger Equation

[Megus]

one-dimensional Schrodinger's Equation

Hi !
I wonder how to solve one-dimensional Schrodinger's Equation :

$$\frac{d^2 \psi (x)}{dx^2}\ = \ -(\frac{2 \pi}{\lambda})^2 \ * \ \psi (x)$$

I've to obtain $$\psi (x)$$ , when
$$-(\frac{2 \pi}{\lambda})^2$$ is known
Can you solve it as an example, when $$-(\frac{2 \pi}{\lambda})^2 = 2$$ ? Please...

 

[speeding electron]

Based on my rather basic knowledge, making
$$- \left( \frac{2 \pi}{ \lambda} \right)^2 = 2$$
would make the psi function an exponential function rather than having the wave characteristics you want.The solution would be:
$$\psi(x) = e^{ \sqrt{2} x } .$$
But you want a periodic wave function. By taking
$$\frac{2 \pi}{\lambda} = \sqrt{2}$$
(i.e. without the minus sign), the solution would be a wave equation and complex:
$$\psi(x) = e^{ \sqrt{2} i x } .$$

 

[quantumdude]

Hi !
I wonder how to solve one-dimensional Schrodinger's Equation :

$$\frac{d^2 \psi (x)}{dx^2}\ = \ -(\frac{2 \pi}{\lambda})^2 \ * \ \psi (x)$$

I've to obtain $$\psi (x)$$ , when
$$-(\frac{2 \pi}{\lambda})^2$$ is known
Can you solve it as an example, when $$-(\frac{2 \pi}{\lambda})^2 = 2$$ ? Please...

You shouldn't commit to a specific value of -(π/λ)². Just define it to be a literal constant, say, -k². Then your differential equation is:

ψ"+k²ψ=0

Surely you have the solution of that one somewhere!

 

[quantumdude]

Based on my rather basic knowledge, making
$$- \left( \frac{2 \pi}{ \lambda} \right)^2 = 2$$
would make the psi function an exponential function rather than having the wave characteristics you want.The solution would be:
$$\psi(x) = e^{ \sqrt{2} x } .$$

No, an exponential function arises when the first derivative is negatively proportional to the wavefunction itself. The problem here involves the second derivative.

But you want a periodic wave function. By taking

The solution to his equation is indeed sinusoidal, but let's let him work it out.

 

[heardie]

In fact $$\frac{{2\pi }} {\lambda } = k$$ when k is the wavenumber, so the k^2 substution can be made even without the knowledge of how it helps solve the DE.

 

[quantumdude]

In fact $$\frac{{2\pi }} {\lambda } = k$$ when k is the wavenumber, so the k^2 substution can be made even without the knowledge of how it helps solve the DE.

Yes, my choice in symbols was not accidental.

 

[Megus]

Is it the correct answer ?:
$$\psi (x) = e^{ikx + \varphi_R + i \varphi_I} = e^{\varphi_R} \cdot e^{i(kx + \varphi_I)} = A(\cos (kx + \varphi_I) + i \sin (kx + \varphi_I))$$

 

[quantumdude]

Is it the correct answer ?:
$$\psi (x) = e^{ikx + \varphi_R + i \varphi_I} = e^{\varphi_R} \cdot e^{i(kx + \varphi_I)} = A(\cos (kx + \varphi_I) + i \sin (kx + \varphi_I))$$

Yes, though you might find it more convenient to replace the constants A and φ_I with A and B as follows:

ψ(x)=Acos(kx)+Bsin(kx),

and applying the boundary conditions from there.

 

[Megus]

Ok - thanks