About wavefunctions of Hydrogen atom

[zhangpujumbo]

Every one knows that wavefunctions are generally complex functions described by three quantum numbers n, l and m, and the number m is included in the form exp(i*m*fai). But here in the following webpage they are all real functions, I'm confused . Can anyone help me?

Thank u in advance!

 

[jtbell]

Uh, what web page?

 

[zhangpujumbo]

Uh, what web page?

sorry, it's missed.

http://www.pci.tu-bs.de/aggericke/PC3e_osv/Kap_IV/H-Wellenfunktion.htm

 

[jtbell]

That page makes a mistake in listing (for example) the $2p_x$ and $2p_y$ wave functions as having m = 1 and -1. They are actually linear combinations of the functions with m = 1 and -1. Recall that

$$\cos \phi = \frac{e^{i \phi} + e^{-i \phi}}{2}$$

$$\sin \phi = \frac{e^{i \phi} - e^{-i \phi}}{2i}$$

If you measure $L_z$ for either of these functions, you get $+ \hbar$ half the time, and $- \hbar$ half the time, randomly.

The $p_x$ and $p_y$ functions are convenient for some purposes because they have lobes along the x and y axes, just like the $p_z$ (m = 0) function has lobes along the z axis.

 

[reilly]

Wave functions can be real; typically this is the case for bound states. (Strictly speaking this holds for the radial function.) Think about harmonic oscillator wave functions -- they are real. Pretty standard stuff.
Regards,
Reilly Atkinson

 

[Hans de Vries]

That page makes a mistake in listing (for example) the $2p_x$ and $2p_y$ wave functions as having m = 1 and -1. They are actually linear combinations of the functions with m = 1 and -1. Recall that

$$\cos \phi = \frac{e^{i \phi} + e^{-i \phi}}{2}$$

$$\sin \phi = \frac{e^{i \phi} - e^{-i \phi}}{2i}$$

If you measure $L_z$ for either of these functions, you get $+ \hbar$ half the time, and $- \hbar$ half the time, randomly.

Yes.

The $p_x$ and $p_y$ functions are convenient for some purposes because they have lobes along the x and y axes, just like the $p_z$ (m = 0) function has lobes along the z axis.

In Cartesean coordinates it's clearer:

$$p_z\ \ \propto\ \ \cos{\theta}\ =\ \frac{z}{r}$$

$$p_x\ \ \propto\ \ \sin{\theta}\cos{\phi}\ =\ \frac{x}{r}$$

$$p_y\ \ \propto\ \ \sin{\theta}\sin{\phi}\ =\ \frac{y}{r}$$

They are all the same.Regards, Hans

 

[zhangpujumbo]

That page makes a mistake in listing (for example) the $2p_x$ and $2p_y$ wave functions as having m = 1 and -1. They are actually linear combinations of the functions with m = 1 and -1. Recall that

$$\cos \phi = \frac{e^{i \phi} + e^{-i \phi}}{2}$$

$$\sin \phi = \frac{e^{i \phi} - e^{-i \phi}}{2i}$$

If you measure $L_z$ for either of these functions, you get $+ \hbar$ half the time, and $- \hbar$ half the time, randomly.

The $p_x$ and $p_y$ functions are convenient for some purposes because they have lobes along the x and y axes, just like the $p_z$ (m = 0) function has lobes along the z axis.

Yes, I agree with your opinion very much!

There must be something wrong.

Thanks a lot

 

[zhangpujumbo]

Wave functions can be real; typically this is the case for bound states. (Strictly speaking this holds for the radial function.) Think about harmonic oscillator wave functions -- they are real. Pretty standard stuff.
Regards,
Reilly Atkinson

I don't mean all wavefunctions must be complex.

But thank u all the same!

 

[zhangpujumbo]

In Cartesean coordinates it's clearer:

$$p_z\ \ \propto\ \ \cos{\theta}\ =\ \frac{z}{r}$$

$$p_x\ \ \propto\ \ \sin{\theta}\cos{\phi}\ =\ \frac{x}{r}$$

$$p_y\ \ \propto\ \ \sin{\theta}\sin{\phi}\ =\ \frac{y}{r}$$

en, it's clearer.

 

[zhangpujumbo]

I don't know how to type mathematical equations here, it's too inconvenient.

How do you do that?

 

[jtbell]

Click here: Introducing LaTeX Math Typesetting

 

[zhangpujumbo]

Click here: Introducing LaTeX Math Typesetting

It seems that all the equations are copied piece by piece, then typying equations will be too laborious a task

Is there a shortcut?

 

[inha]

Not really. But LaTex is easy once you get past the initial shock.

 

[zhangpujumbo]

Not really. But LaTex is easy once you get past the initial shock.

I think a compact software like mathtype will help greatly.