Linear combination of states

In summary, the linear combination of states |j,m + 1/2> = a|lm,+> + b|lm,-> is commonly used in quantum mechanics textbooks to explain the addition of spin 1/2 and orbital angular momentum. The left hand side is equal to the right hand side through the use of ladder operators, starting with the total state |J,MJ>=|L,ML>*|1/2,1/2> and working down. There is a parity change when reaching J=1/2.
  • #1

positron98

The following linear combination of states is considered in almost all quantum mechanics textbooks when they try to explain the addition of spin 1/2 and orbital angular momentum. The thing I don't understand is how the left hand side is equal to the right. Please, if you can, explain how.

|j,m + 1/2> = a|lm,+> + b|lm,-> where |+> is spin up and |-> is spin down.


Thanks,

Sam
 
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  • #2
Write the spin and orbital states separately, and the total as a direct product of the two, as follows:

|J,MJ>=|L,ML>*|1/2,1/2>

Then use the ladder operators to work your way down.

Example:

L=1, S=1/2

|3/2,3/2>=|1,1>*|1/2,1/2>

Apply the ladder operator J-=L-+S-, noting that on the RHS the operator only acts on its respective ket.

J-|3/2,3/2>=(L-|1,1>)*|1/2,1/2>+|1,1>*(S-|1/2,1/2>)

and keep working down.

Try to work out the combination for |J=3/2,MJ=1/2>. If you get stuck, post what you came up with and I'll help you through it.
 
  • #3
What the hell, I'm feeling ambitious!

J-|3/2,3/2>=(L-|1,1>)*|1/2,1/2>+|1,1>*(S-|1/2,1/2>)

RHS:
J-|3/2,3/2>=31/2(hbar)|3/2,1,2>

LHS:
(L-|1,1>)*|1/2,1/2>+|1,1>*(S-|1/2,1/2>)=
21/2(hbar)|1,0>*|1/2,1/2>+(hbar)|1,1>*|1/2,-1/2>

Putting them together and solving for |3/2,1/2> yields:

|3/2,1/2>=(2/3)1/2|1,0>*|1/2,1/2>+(1/3)1/2|1,1>*|1/2,-1/2>

Try the rest, and let me know if you need help.

Remember: Once you get to J=1/2, you will have a parity change. See me if you need help on that, too.
 
  • #4
Thanks Tom for your time and help.

Sam
 

1. What is a linear combination of states?

A linear combination of states refers to a mathematical operation in which multiple quantum states are combined using coefficients and operators to create a new composite state. This is commonly used in quantum mechanics to describe the state of a system.

2. How is a linear combination of states represented?

A linear combination of states is typically represented using a linear combination of basis states, where each basis state has a corresponding coefficient. The coefficients are multiplied by the basis states and then added together to create the composite state.

3. What is the significance of a linear combination of states in quantum mechanics?

In quantum mechanics, a linear combination of states is important because it allows us to describe the state of a system in terms of simpler, known states. This makes it easier to analyze and understand complex quantum systems and their behavior.

4. How does the principle of superposition relate to linear combinations of states?

The principle of superposition states that a physical system can exist in multiple states at the same time. This is similar to a linear combination of states, where a composite state is created by adding together multiple individual states. This principle is fundamental to understanding quantum mechanics.

5. Can linear combinations of states be used to describe classical systems?

No, linear combinations of states are specific to quantum systems and cannot be applied to classical systems. This is because classical systems do not exhibit the same principles of superposition and uncertainty that are fundamental to quantum mechanics.

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