Angular momentum of an electron

In summary, the conversation discusses the possibility of an electron in a hydrogen atom being in a state that is a superposition of energy eigenstates and having different expectation values for L_x and L_y. It is concluded that this is possible and the theory treats x and y equally when z is chosen as the reference axis. However, a specific state may not have symmetry about the z-axis, resulting in different expectation values for L_x and L_y. The conversation also includes a question about an assignment involving a hydrogen atom with a specific state and the calculation of expectation values for L_x and L_y.
  • #1
broegger
257
0
Hi,

Say you have an electron in the hydrogen atom. Can this electron be in a state that is a superposition of the usual energyeigenstates (which are also eigenstates of L_z) AND have different expectation values for L_x and L_y? Are x and y symmetric in the sense that their expectation values are always equal in this situation?
 
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  • #2
You can't measure the expectation value of L_z and L_y simultaneously cause of the uncertainty. You can only measure total angular momentum and L_y or total angu. mom. and L_z precisely.
 
  • #3
You can measure them all (5) with arbitrary precision,if the quantum system is in a state with [itex] l=0 [/itex].

Daniel.
 
  • #4
Kruger said:
You can't measure the expectation value of L_z and L_y simultaneously cause of the uncertainty. You can only measure total angular momentum and L_y or total angu. mom. and L_z precisely.

No no, I think you misunderstand. The EXPECTATION value is not the result of a single measurement. This is what I mean: Can you have a particle in a state, which is a superposition of eigenstates of L_z, but which - at some point in time (say t=0) - has different EXPECTATION values for L_x and L_y.

The reason I'm asking is that I found just that in my exam the other day. The assignment was this:

In this problem we consider a hydrogen atom whose normalized stationary states is denoted [tex]|nlm\rangle[/tex], where n is the main quantum number and l and m is the quantum numbers belonging to L^2 and L_z, respectively. Consider a hydrogen atom described by the state:
[tex]|\psi\rangle=\frac1{\sqrt2}(|210\rangle + |211\rangle)[/tex]​

Answer these questions:

1. Blah blah
2. Blah blah
3. Blah blah
4. Calculate the expectation values of L_x and L_y in the state |psi>.

I got different answers for <L_x> and <L_y> (h/sqrt(2) and 0, respectively). Is this possible? When you choose z as reference axis the theory seems to treat x and y equally.
 
  • #5
There's no problem whatsoever.The two operators are different (see their expressions as a function of the ladder operators),so it's obvious to get different expectation values.

Daniel.
 
  • #6
Ok, thanks.
 
  • #7
broegger said:
When you choose z as reference axis the theory seems to treat x and y equally.

That's right, but your specific state doesn't treat x and y equally. So, as dextercioby already pointed, there's no contradiction if you find different expectation values.
 
  • #8
Choosing z as the reference axis doesn't necessarily treat x and y equally, it's
just a convention. Only m=0 states have symmetry about the z-axis. So <L_x> = <L_y> (= 0 I think) whenever m= 0, but since [tex] |211\rangle [/tex] is part of
the wavefunction, it doesn't have the symmetry.
 

What is angular momentum?

Angular momentum is a measure of the rotational motion of an object, or in the case of an electron, its spinning motion around its own axis.

How is angular momentum of an electron related to its energy level?

The angular momentum of an electron is directly related to its energy level. As an electron moves to a higher energy level, its angular momentum increases.

What is the formula for calculating the angular momentum of an electron?

The formula for calculating the angular momentum of an electron is L = mvr, where L is the angular momentum, m is the mass of the electron, v is its velocity, and r is the distance of the electron from the nucleus.

Why do electrons have quantized angular momentum?

Electrons have quantized angular momentum because they exist in discrete energy levels, and their angular momentum can only take on certain values in these levels. This is due to the wave-like nature of electrons, as described by quantum mechanics.

How does the angular momentum of an electron affect its behavior in an atom?

The angular momentum of an electron plays a significant role in determining the behavior of electrons in an atom. It affects their energy levels, orbital shapes, and interactions with other particles, ultimately determining the chemical and physical properties of elements.

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