What does it mean to satisfy the Schrodinger equation?

In summary, the conversation discusses the concept of "satisfying" the Schrodinger equation and how it applies to the 2p wave functions of the hydrogen atom. The conversation also includes a discussion on how to show that the 2p wave functions satisfy the radial Schrodinger equation and provides an example calculation using the Bohr radius.
  • #1
warfreak131
188
0
What does it mean to "satisfy" the Schrodinger equation?

Homework Statement



Show that the 2p wave functions of the hydrogen atom satisfy the radial Schrodinger eq.

One of the radial equations for the 2p state is [tex]\frac{1}{\sqrt{96 \pi a^{3}}} \frac{r}{a} e^{\frac{-r}{2a}}[/tex]

Homework Equations


The Attempt at a Solution



[tex][\frac{-\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}(r^2\frac{d}{dr})+\frac{l(l+1)\hbar^2}{2mr^2}+V(r)]R=ER[/tex]

I took the derivative with respect to r, and followed all the subsequent derivatives, and the answer is really messy. What exactly am I looking for when something "satisfies" the equation?
 
Physics news on Phys.org
  • #2


You're looking to see if the operator on the left, when acting upon the wave function (or in this case part of it), returns a constant times that same wave function.

So if I had something like [tex]f(x)=e^{-3x/5}[/tex] and were asked if f(x) satisfies

[tex]{{df}\over{dx}} = Af(x)[/tex].

What you get when you do the derivative of f(x) is the -3/5 times the function again. Thus, f(x) satisfies the equation with A = -3/5. Same idea here
 
  • #3


Ok, I see. So what should I do since all the derivates and sums don't come out to a nice constant?
 
  • #4


Check your work :P Or show us your work
 
  • #5


okay, but quick question first
[tex]\frac{d}{dr}(r^2\frac{d}{ dr})[/tex]

is this line asking you to take the derivative of R, multiply it by r^2, and then take the derivative of the resulting equation?
 
  • #6


warfreak131 said:
okay, but quick question first
[tex]\frac{d}{dr}(r^2\frac{d}{ dr})[/tex]

is this line asking you to take the derivative of R, multiply it by r^2, and then take the derivative of the resulting equation?

Yup!
 
  • #7


okay, i used mathematica, i created functions for each term, did all the necessary derivatives and additions, and what i got was

(2 a^2 e^2 m - 8 a h^2 pi epsilon + h^2 pi r epsilon)/(8 a^2 m pi r epsilon)

this answer was close with the exception of r, everything else is a constant
 

Attachments

  • Untitled.png
    Untitled.png
    8.2 KB · Views: 871
  • #8


Express the Bohr radius in terms of the other constants. You should find the first two terms in the numerator will cancel.
 
  • #9


vela said:
Express the Bohr radius in terms of the other constants. You should find the first two terms in the numerator will cancel.

ill try that, thanks
 

What is the Schrodinger equation?

The Schrodinger equation is a mathematical equation that describes the behavior of particles at the quantum level. It is a fundamental equation in quantum mechanics and is used to calculate the wave function of a particle at any point in time.

Why is satisfying the Schrodinger equation important?

Satisfying the Schrodinger equation is important because it allows us to accurately describe the behavior of particles at the quantum level. It is a crucial step in understanding and predicting the behavior of atoms, molecules, and other particles.

How is the Schrodinger equation satisfied?

The Schrodinger equation is satisfied when the calculated wave function of a particle matches the observed behavior of the particle. This means that the wave function accurately predicts the location, energy, and other properties of the particle.

What are the implications of satisfying the Schrodinger equation?

Satisfying the Schrodinger equation has many implications in the field of quantum mechanics. It allows us to make accurate predictions about the behavior of particles, which has led to advancements in technology such as transistors and lasers. It also helps us understand the nature of reality at the smallest scales.

Are there any limitations to satisfying the Schrodinger equation?

While the Schrodinger equation is a powerful tool, it does have its limitations. It cannot accurately describe the behavior of particles moving at high speeds or in strong gravitational fields. Additionally, it does not take into account the effects of relativity, which becomes important at these extreme conditions.

Similar threads

Replies
2
Views
961
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
920
  • Advanced Physics Homework Help
Replies
1
Views
588
  • Advanced Physics Homework Help
Replies
30
Views
1K
  • Advanced Physics Homework Help
Replies
14
Views
954
  • Advanced Physics Homework Help
Replies
1
Views
859
  • Advanced Physics Homework Help
Replies
16
Views
859
  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
0
Views
344
Back
Top