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Two electrons have ##l_1=1## and ##l_2=3##, what are L and S

  1. Oct 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Two electrons in helium have ##l_1=1## and ##l_2=3##. What are the values of ##L## and ##S##? From this, deduce the possible values of ##J## and find how many quantum states this excited state of helium can occupy.

    2. Relevant equations


    3. The attempt at a solution
    For ##L## the allowed values are given by ##L= \hbar m##, so for ##l_1=1## ##L = hbar##, ##0## and ##L=-\hbar##.
    For ##l_2##,
    ##L = -3\hbar##, ##-2\hbar##, ##-\hbar##, ##0##, ##\hbar##, ##2\hbar## and ##L=3\hbar##.

    For ##S## the allowed values are again the eigenvalues which are ##\hbar m_s##, ##m_s## runs from ##-s## to ##s## in integer steps. Electrons have ##s=\frac{1}{2}##, so
    ##S = -\frac{1}{2}\hbar## or ##S = \frac{1}{2} \hbar##.

    I'm not sure how to work out ##J##, in my lecture notes it says for a single electron ##j = l \pm \frac{1}{2}## but doesn't say how that changes for multiple electrons. ##J=\hbar m_j## where ##m_j## runs from ##-j## to ##j##.

    For the last part, I think for every combination of ##L##, ##J## and ##S## there are ##2J+1## quantum states. So I think the main question I have is: how can I work out the allowed values of ##J##? Can I work them out individually for each electron using ##j = l \pm \frac{1}{2}## and then add them?
     
  2. jcsd
  3. Oct 16, 2016 #2

    kuruman

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    It looks like you are confusing L with Lz. When L = 1, Lz can take values ## \pm \hbar## and zero.
    It also looks like you are not sure about the rules for adding angular momenta. You are given l1 = 1 and l2 = 3. You are asked to find the possible values for L defined as L = l1 + l2. Does this look familiar?
     
  4. Oct 16, 2016 #3
    Well the only things in my lecture noted are ##L^2## and ##L_z##, so I assumed the ##L## I was being asked about was ##L_z##. Ok, so I should be using ##L = l_1+l_2##? I was also using the definition for ##S_z##, does a similar thing apply there? Should I be calculating ##S = s_1+s_2##? But I think ##s_1## and ##s_2## can both have two values, ##\frac{1}{2}## and ##-\frac{1}{2}##, so would I need to account for that?
     
  5. Oct 16, 2016 #4

    kuruman

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    Yes, you should be using ##L=l_1+l_2##. And yes, a similar thing applies to ##S=s_1+s_2##. No, ##s_1## and ##s_2## cannot have the values you suggest. Just like the orbital angular momentum, ##s_{1z}= \pm \frac{\hbar}{2}## and similarly for ##s_{2z}##. You need to add angular momenta three times:
    1. ##L=l_1+l_2##
    2. ##S=s_1+s_2##
    3. ##J=L+S##
     
  6. Oct 16, 2016 #5
    Is ##s_{1z}=s_1##? Or are they different, different orientation or something?
     
  7. Oct 16, 2016 #6

    kuruman

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    They are different. Think of ## s_1=\frac{1}{2}## as a vector. It's a quantum vector the component of which in the z-direction is denoted by ##s_{1z}## and can have only two values, ##+\frac{\hbar}{2}## and ##-\frac{\hbar}{2} ##. The same holds for ##s_2##. You are looking for the sum ##S=s_1+s_2##. The sum of two quantum vectors must also be a quantum vector. This means that it can have an integer or half-integer value. Do you remember learning about this? You may wish to look at this Wikipedia entry https://en.wikipedia.org/wiki/Total_angular_momentum_quantum_number.
     
  8. Oct 17, 2016 #7
    Yes, it definitely rings a bell. So this link seems to suggest that since ##S^2 = \hbar^2 s(s+1)## then I would square root this to find ##S##:
    https://en.wikipedia.org/wiki/Spin_quantum_number#Electron_spin.
    The bit I was looking at is section 3.

    ##s=\frac{1}{2}## for electrons so then would I add the values from this equation? I'd get ##S = 2\hbar\sqrt{\frac{3}{4}}##. And then it also says this is analagous to how you'd calculate ##L## but if I repeated the same thing for ##L## as for ##S## that would be a different answer to just adding ##l_1## and ##l_2##. I'm having trouble working out from different texts what they are using L and S and all these capital and little letters for, because it feels like they're being used for different things, or I'm just getting very confused. And so I can't identify how these equations I'm finding relate to the numbers in my question.

    The other possibliity would be ##S=1## and ##L=4##...
     
    Last edited: Oct 17, 2016
  9. Oct 17, 2016 #8

    kuruman

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    OK, here are the rules for adding angular momenta. They apply to all angular momentum quantities regardless of the symbol you use. Say you add ##l_1## and ##l_2## to get ##L##. The possible values for ##L## start at ##l_1+l_2## and go down in decrements of 1 all the way to ##|l_2-l_1|##. For example, if you were to add l1 = 4 and l2=7 to get L, the possible values are L = 11, 10, 9, 8, 7, 6, 5, 4, 3. If you were to add l=4 and s=3/2 to get j, the possible values are j = 11/2, 9/2, 7/2, 5/2. Do you see how it works? If yes, apply it to your problem and consult with post#4 on how to add things.
     
  10. Oct 18, 2016 #9
    That makes sense, I'll apply that to my problem! I was having real trouble deciphering my lecture notes. Thank you for your help and patience, I really appreciate it! :)
     
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