Angular dependence in QM: Why is it present in the hydrogen atom?

In summary, the angular dependence of eigenstates in the hydrogen atom is due to the fact that the solutions to the Schrodinger equation are not necessarily spherically symmetric.
  • #1
Repetit
128
2
When calculating eigenstates in the hydrogen atom one finds plenty of eigenstates with angular dependence. The s orbitals are spherically symmetric, but an orbital like 2p is not, there is some angular dependence through the spherical harmonics. But why is there angular dependence? It is a totally spherically symmetric problem so how can there be? Certainly, starting from the nucleus and going out in one direction should not be any different from going out in any other direction. How is the angular dependence to be interpreted?
 
Physics news on Phys.org
  • #2
Recall that most planets have elliptical orbits while undergoing a central force, the sun's gravity.
Regards,
Reilly Atkinson
 
  • #3
Repetit said:
When calculating eigenstates in the hydrogen atom one finds plenty of eigenstates with angular dependence. The s orbitals are spherically symmetric, but an orbital like 2p is not, there is some angular dependence through the spherical harmonics. But why is there angular dependence? It is a totally spherically symmetric problem so how can there be? Certainly, starting from the nucleus and going out in one direction should not be any different from going out in any other direction. How is the angular dependence to be interpreted?
Consider a completely classical situation
Even if the force is spherically symmetric, the initial conditions of the motion don't have to be!

If the force is a central force, the angular momentum will be conserved. But the motion is not necessarily spherically symmetric since you can choose whatever initial conditions you want!

Consider tossing a comet at some arbitary direction in the solar system (and assume it doe snot have enough energy to escape the potential). It will move following an ellipse which is obviosuly not spherically symmetric.

Again the key point is that you must consider both the symmetry of the force and the symmetry of the initial conditions.
 
  • #4
rephrasing the last post in "quantum terms", even thought the potential which appears in the schorodinger equation is spherically symmetric the solutions to the schrodinger equation are not necessarily spherically symmetric.
 
  • #5
kdv said:
Consider a completely classical situation
Even if the force is spherically symmetric, the initial conditions of the motion don't have to be!

If the force is a central force, the angular momentum will be conserved. But the motion is not necessarily spherically symmetric since you can choose whatever initial conditions you want!

Consider tossing a comet at some arbitary direction in the solar system (and assume it doe snot have enough energy to escape the potential). It will move following an ellipse which is obviosuly not spherically symmetric.

Again the key point is that you must consider both the symmetry of the force and the symmetry of the initial conditions.


Great! The solutions of an aquation may have less symmetry than the equation.
 
  • #6
jiadong said:
Great! The solutions of an aquation may have less symmetry than the equation.

Welcome to PF, Jiadong. Just a heads-up, though - you might want to check the date (upper-left corner of the post) before responding; you've answered a few that are a bit old.
 

1. What is angular dependence in quantum mechanics?

Angular dependence in quantum mechanics refers to the relationship between the direction of a particle's motion and its energy. It is a fundamental concept in quantum mechanics and is described by the angular momentum operator.

2. How is angular dependence related to the uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know both the position and momentum of a particle with absolute certainty. This also applies to angular dependence, as measuring the direction of a particle's motion will inevitably affect its energy and vice versa.

3. What is the importance of angular dependence in quantum mechanics?

Angular dependence is crucial in understanding the behavior of particles on a microscopic scale. It plays a significant role in determining the energy levels of atoms and molecules, as well as the behavior of particles in electromagnetic fields.

4. How does angular dependence affect the properties of atoms and molecules?

The angular dependence of particles in atoms and molecules is responsible for many of their unique properties. For example, it determines the shape of atomic and molecular orbitals, which in turn affects their chemical and physical properties.

5. Can angular dependence be experimentally observed?

Yes, angular dependence can be observed through various experiments in quantum mechanics. For example, the double-slit experiment demonstrates the wave-like nature of particles and their angular dependence, as the interference pattern changes depending on the direction of the particles' motion.

Similar threads

Replies
2
Views
753
  • Quantum Physics
Replies
6
Views
1K
  • Quantum Physics
Replies
2
Views
284
Replies
17
Views
1K
  • Quantum Physics
Replies
1
Views
842
Replies
9
Views
1K
  • Quantum Physics
Replies
11
Views
1K
Replies
1
Views
1K
Replies
18
Views
1K
  • Quantum Physics
Replies
2
Views
712
Back
Top