Addition of angular momentum MADE EASY!!! (with example for two s=1/2 states)

[@NickHeumannUniversity]

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[@itsfrancyo1942]

I've a quantum mechanics exam tomorrow, I spent a lot of time trying to understand the addition of angular momentum and searching for good lecture material online but it didn't help. Then I came across your video and, man, you saved me.

[@NickHeumannUniversity]

Glad I could help! Best of luck in your exam!

[@pedroafonso8384]

thank you so much! really clear

[@noelwass4738]

Wonderfully explained. On looking at this closely the only thing not clear to me is why the lowering operator on the tensor product of the two spin 1/2 states is given by the sum of the lowering operators on each component of the tensor product. But that is probably just my ignorance of lowering operator on the tensor product.

[@aaaaaaaaaaaaabaab]

Thank you for this!!!

[@shireensultana9738]

REALLY HELPFUL

[@Gabriel_Marin]

Great video!

[@brenthughes8101]

What software do you use for these wonderful videos?

[@koticcafc]

you're awesome

[@Artizz25]

Great explanation. I have a question related to the addition of angular momenta. If there is an electron with l=2 there are common eigenkets of L^2, S^2, S3 and J^2, L^2 , S^2. How can we express one in terms of the other if J=L+S. Example |5/2, 2, 1/2 ; 5/2> = |2, 2; 1/2, 1/2>. How can I argue this?

[@NickHeumannUniversity]

Hi! I'm glad the video was useful to you! I'm afraid that I don't quite understand your question, though. Could you elaborate further please?

[@Artizz25]

@@NickHeumannUniversity thank you for replying. I am learning quantum mechanics and I came across this problem. "Suppose that an electron is in a state of orbital angular momentum l=2. An orthonormal basis for the states is given by simultaneous eigenstates of L^2, L3, S^2 and S3 as |l, ml ; s, ms>. Alfernatively we can choose an orthonormal basis as simultaneous eigenstates of J^2, L^2, S^2 and J3 with J=L+S as |j, l, s; mj>. Argue that |5/2, 2, 1/2 ; 5/2> = |2, 2; 1/2, 1/2>.

[@NickHeumannUniversity]

@@Artizz25 I don't know what context that problem is in, but it would seem that you have to add the states using addition of angular momentum (adding 2 + 1/2), and you would have to show that the result would be the quantum numbers from the other case. Hope it helps!

[@raufabid1925]

Beautifully explained!! But I have a question?

[@raufabid1925]

Initially, for [1 1> or state 1 1> you have applied lowering operator. At last for 0 0> you have chosed a^2+b^2=1 to determine the co-officent. The question is can we use both ways to determine co-officents???

[@NickHeumannUniversity]

We cannot use both procedures in any case. For example, for0 0> we cannot use raising or lowering operators, since there are no other states for that same value of "l". That is why we had to use the inner product. Hope it helps!

[@raufabid1925]

What do you mean by same value of l? Kindl plz elaborate..

[@NickHeumannUniversity]

@@raufabid1925 whenever you have a state |n,l,m> You have 2l+1 values of m for each value of l. This is true for states |j_n.j_l,j_s> which arise after adding up l+s. Whenever you apply the raising operator, you go from m to m+1, and when you apply the lowering operator, you go from m to m-1. All of this is true for J states too, but the names change. so for example yoou have 2j+1 values of m for each value of j

[@raufabid1925]

@@NickHeumannUniversity thx bro.

[@lorenzo7571]

why are both spins 1/2 in the ket (0,0)? Shouldn´t they have opposite signs?

[@NickHeumannUniversity]

Mind the notation that I used. in my notation, in the ket (0,0) i have (s1,s2,m1,m2), so (1/2,1,2;1/2,-1/2) means s1=1/2=s2 (both particles have spin 1/2) but their m values can change (they can be in state 1/2 or -1/2). I hope this clears it up

[@koticcafc]

where did you learn that?

[@brenthughes8101]

How can I contact you to hire you to instruct me in using the type of software you use?

[@NickHeumannUniversity]

Send me an email to [e-mail]

[@themorrigan3673]

I cant seem ti understand why there is a Stotal = 0 ? arent they both +1/2 particle

[@NickHeumannUniversity]

Yes, but think of it as the "absolute value". You can have 1/2 spin in one direction, which we call +1/2, or you can have it pointing in the opposite direction, which we call - 1/2. Both still have a magnitude of 1/2,put have a difference in direction. So if you have two particle with spin 1/2,you can have them pointing in the same direction 1/2+1/2=1,in opposite direction, either 1/2+-1/2 =0 or - 1/2 +1/2=0 (same resulta in both cases). Finally, you can have - 1/2+-1/2=-1. Hope this helps

[@themorrigan3673]

@@NickHeumannUniversity Ohh ok but why isnt the Stotal = -1 included? Is it because of the "absolute value", if so why only the absolute value?

[@NickHeumannUniversity]

@@themorrigan3673 im not sure to which part of the video you refer, but when you add quantum numbers, you get New quantum numbers that obey the normal rules. So if after adding l1 and l2 you get l_total=1 (since the l for spin 1/2 is 1/2). So if l_total is 1,then the values of m_total goe from +l to - l in steps of 1 so you get 1,0,-1. Thus, l can be 1 and 0, m can be 1,0,-1 Hope this makes it clearer