The wave function of hydrogen

In summary, the conversation discusses the angular portion of the wavefunction of hydrogen and the corresponding relationships between different functions such as dxy, dxz, dyz, dz2, and dx2-y2. The use of cartesian coordinates to describe the geometry of the wavefunction is also mentioned. The convention for assigning values of m to different functions is also discussed, but it is noted that there are no strict relationships between the two. References for further reading on this topic are also provided.
  • #1
einstein1921
76
0
HI,everyone.I have a problem. the angular portion of wavefunction of hydrogen,like 3d.
n=3,l=2,so m=2,1,0,-1,-2.I read some books that say dxy,dxz,dyz,dz2,dx2-y2,so what the
corresponding Relation between them. for example,dz2 corresponding what ?m=0?? and why?
any help will be highly appreciated!
 
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  • #2
You are correct. The cartesian coordinates subscripts usually indicate nodal planes. For example, the p_z function (l=1, m=0)can be written as:

[tex]

\psi = \frac{1}{\sqrt{32 \pi}} \left( \frac{Z}{a_0} \right)^{5/2} z e^{\frac{-Zr}{2 a_0}}

[/tex]

Where cartesian and polar coordinates are mixed together. However, when written in this form, it is easy to see that whenever z=0 then psi also = 0, which means that the xy plane is a nodal plane.

As for d functions, it becomes more complicated. I think that some of them will be complex functions, and you'll need to take linear combinations of them in order to get real functions that you can plot. The d_(x^2-y^2) function is a linear combination of the complex m=+2 and m=-2 functions. This will yield:

[tex]
\psi = \frac{1}{81\sqrt{ 2\pi}} \left( \frac{Z}{a_0} \right)^{7/2} e^{\frac{-Zr}{3 a_0}} (x^2 - y^2)[/tex]

So, in a way, this notation with the cartesian coordinates describes the geometry of the wavefunction.
 
  • #3
Amok said:
You are correct. The cartesian coordinates subscripts usually indicate nodal planes. For example, the p_z function (l=1, m=0)can be written as:

[tex]

\psi = \frac{1}{\sqrt{32 \pi}} \left( \frac{Z}{a_0} \right)^{5/2} z e^{\frac{-Zr}{2 a_0}}

[/tex]

Where cartesian and polar coordinates are mixed together. However, when written in this form, it is easy to see that whenever z=0 then psi also = 0, which means that the xy plane is a nodal plane.

As for d functions, it becomes more complicated. I think that some of them will be complex functions, and you'll need to take linear combinations of them in order to get real functions that you can plot. The d_(x^2-y^2) function is a linear combination of the complex m=+2 and m=-2 functions. This will yield:

[tex]
\psi = \frac{1}{81\sqrt{ 2\pi}} \left( \frac{Z}{a_0} \right)^{7/2} e^{\frac{-Zr}{3 a_0}} (x^2 - y^2)[/tex]

So, in a way, this notation with the cartesian coordinates describes the geometry of the wavefunction.

Thank you,Amok.your answer is very helpful to me.would you please tell me that p_x corresponds to m=1 or m=-1 and why?
again thanks.
 
  • #4
einstein1921 said:
Thank you,Amok.your answer is very helpful to me.would you please tell me that p_x corresponds to m=1 or m=-1 and why?
again thanks.
It is more of a convention thing than anything; there are no strict relationships between the two. You can define whatever.

That being said, the most common convention in quantum chemistry has the real solid harmonics beeing approximated like this in terms of cartesians:

AngMom L = 0
S(0,m=+0) = + 1.00000
AngMom L = 1
S(1,m=-1) = + 1.00000 y
S(1,m=+0) = + 1.00000 z
S(1,m=+1) = + 1.00000 x
AngMom L = 2
S(2,m=-2) = + 1.73205 x y
S(2,m=-1) = + 1.73205 y z
S(2,m=+0) = + 1.00000 z^2 - 0.50000 y^2 - 0.50000 x^2
S(2,m=+1) = + 1.73205 x z
S(2,m=+2) = - 0.86603 y^2 + 0.86603 x^2
AngMom L = 3
S(3,m=-3) = - 0.79057 y^3 + 2.37171 x^2 y
S(3,m=-2) = + 3.87298 x y z
S(3,m=-1) = + 2.44949 y z^2 - 0.61237 y^3 - 0.61237 x^2 y
S(3,m=+0) = + 1.00000 z^3 - 1.50000 y^2 z - 1.50000 x^2 z
S(3,m=+1) = + 2.44949 x z^2 - 0.61237 x y^2 - 0.61237 x^3
S(3,m=+2) = - 1.93649 y^2 z + 1.93649 x^2 z
S(3,m=+3) = - 2.37171 x y^2 + 0.79057 x^3

This is what you will usually get when using "spherical" basis functions in a chemical electronic structure program, and correspondingly also what most plots and chemistry books refer to.
 
  • #5
Exact same thing as p_z, except you the nodal plane will be yz, instead of xy. The fact that it is assigned to m=+1 is just a convention. Check this out:

http://en.wikipedia.org/wiki/Atomic_orbital#Orbitals_table

Also, if you want more details, check out QM textbooks. I use the Cohen-Tanoudji (because I'm a French speaker, but is also available in English). What you are asking about is explained in chapter VII, complement E (in some detail, but they give references as well). I think that if you look into good quantum chemistry books, you're bound to find explanations about this, since chemists use these concepts in order to explain chemical phenomena on a daily basis.
 

FAQ: The wave function of hydrogen

What is the wave function of hydrogen?

The wave function of hydrogen is a mathematical representation of the quantum state of a hydrogen atom. It describes the probability of finding the electron in a certain location at a given time.

Why is the wave function of hydrogen important?

The wave function of hydrogen is important because it allows us to understand the behavior and properties of the electron in the hydrogen atom. It is also a fundamental concept in quantum mechanics and is used to calculate various physical quantities, such as energy levels and transition probabilities.

What does the wave function of hydrogen tell us about the atom?

The wave function of hydrogen tells us about the energy levels of the electron, the probability of finding the electron in a certain location, and the overall behavior of the electron in the atom. It also provides information about the shape and orientation of the electron's orbit around the nucleus.

How is the wave function of hydrogen calculated?

The wave function of hydrogen is calculated using Schrödinger's equation, which is a mathematical equation in quantum mechanics that describes the wave-like behavior of particles. The equation takes into account the potential energy of the electron and the kinetic energy of the electron.

What are the possible values of the wave function of hydrogen?

The possible values of the wave function of hydrogen depend on the quantum numbers associated with the electron, such as the principal quantum number, angular momentum quantum number, and magnetic quantum number. These values determine the shape, size, and orientation of the electron's orbital around the nucleus.

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