Dimension of Hilbert space (quantum mechanics)

In summary, the dimension of the Hilbert space to describe all states with the quantum numbers n = l = 1 and s = 1/2 is 6, as it is dependent on the combined angular momentum J = L + S.
  • #1
Haye
15
2

Homework Statement


Consider the states with the quantum numbers n = l = 1 and s = 1/2
Let
J = L + S
What is the dimension of the Hilbert space to describe all states with these
quantum numbers?

Homework Equations




The Attempt at a Solution


I believe the dimension of the Hilbert space is dependent on the number of base vectors.

If I am correct, the 'spin-angle functions' would give me the number of bases vector, and thus the dimension of the Hilbert-space.

This is dependent on the quantum numbers s and ms, l and ml and j and mj.
s=1/2, so ms=-1/2 , +1/2
l=1, so ml = -1, 0, +1
j= 1/2 or 3/2, so mj = -1/2 , 1/2 OR -3/2, -1/2, +1/2, +3/2

For j=1/2 you get 2*3*2=12 different combinations, and for j=3/2 it's 2*3*4=24 different combinations.
This gives 36 different base vectors, and therefore the dimension of Hilbert space would be 36?

I have no idea how to go from here. I have the book "introduction to quantum mechanics" by D.J. Griffiths.

If someone could help me out, it would be greatly appreciated.
 
Last edited:
Physics news on Phys.org
  • #2
Haye said:

Homework Statement


Consider the states with the quantum numbers n = l = 1 and s = 1
Let
J = L + S
What is the dimension of the Hilbert space to describe all states with these
quantum numbers?

Homework Equations




The Attempt at a Solution


I believe the dimension of the Hilbert space is dependent on the number of base vectors.
s=1/2 gives two of these base vectors, with m=±1/2.

I have no idea how to go from here. I have the book "introduction to quantum mechanics" by D.J. Griffiths.

If someone could help me out, it would be greatly appreciated.
Is that supposed to be S=1 or S=1/2? The problem statement you gave said S=1, but you seem to be assuming S=1/2.

Review the addition of angular momenta in your book.
 
  • Like
Likes 1 person
  • #3
s should be 1/2, sorry for that. Thanks for pointing me in the right direction, I'm a lot closer to the answer now (I think so, atleast).

If I am correct, the 'spin-angle functions' would give me the number of bases vector, and thus the dimension of the Hilbert-space.

This is dependent on the quantum numbers s and ms, l and m and j and mj.
s=1/2, so ms=-1/2 , +1/2
l=1, so ml = -1, 0, +1
j= 1/2 or 3/2, so mj = -1/2 , 1/2 OR -3/2, -1/2, +1/2, +3/2

For j=1/2 you get 2*3*2=12 different combinations, and for j=3/2 it's 2*3*4=24 different combinations.
This gives 36 different base vectors, and therefore the dimension of Hilbert space would be 36?

I am completely unsure if my logic is correct, and I would greatly appreciate it if someone could tell me if I am wrong or not.

vela, thank you so much for pointing me in the right direction at least. I feel a lot less lost than I was.
 
  • #4
You're over-counting. The Hilbert space is spanned by either basis vectors ##\lvert l, m_l; s, m_s \rangle## or basis vectors ##\lvert j, m_j \rangle##. In other words, you can look at the individual angular momenta or you can look at the combined angular momenta, but not both at the same time. Also, take the state ##\lvert m_l = 1, m_s = +1/2 \rangle##, for instance. Its the same state as ##\lvert m_j = 3/2 \rangle##.

If you count either set up, you'll see you have 6 basis vectors, so the dimension is 6.
 
  • Like
Likes 1 person
  • #5
Ah, that makes much more sense, J=L+S ofcourse. I understand it know, thank you so much for your help (:
 

1. What is a Hilbert space in quantum mechanics?

A Hilbert space in quantum mechanics is a mathematical concept that represents the state space of a quantum system. It is a complex vector space with an inner product that allows for the calculation of probabilities and expectation values of quantum states.

2. How is the dimension of a Hilbert space determined?

The dimension of a Hilbert space is determined by the number of basis states that span the space. In quantum mechanics, these basis states represent all possible states that a system can be in. The dimension is equal to the number of basis states, which can be finite or infinite.

3. Why is the dimension of a Hilbert space important in quantum mechanics?

The dimension of a Hilbert space is important because it determines the number of degrees of freedom in a quantum system. This means that the more basis states a system has, the more complex its behavior and interactions can be. The dimension also affects the types of operations and measurements that can be performed on a system.

4. How does the dimension of a Hilbert space relate to superposition?

The dimension of a Hilbert space is directly related to the concept of superposition in quantum mechanics. Superposition occurs when a quantum system is in a combination of multiple states at the same time. The dimension of the Hilbert space determines the number of states that can be superimposed, as well as the probability amplitudes for each state.

5. Can the dimension of a Hilbert space change?

The dimension of a Hilbert space is typically fixed for a given system. However, it is possible for the dimension to change if the system undergoes a phase transition or a measurement collapses the state into a different basis. In these cases, the dimension of the Hilbert space may increase or decrease, but it will still represent the state space of the system.

Similar threads

  • Quantum Physics
2
Replies
61
Views
1K
  • Quantum Physics
2
Replies
35
Views
393
  • Advanced Physics Homework Help
Replies
17
Views
1K
  • Quantum Physics
Replies
7
Views
1K
Replies
1
Views
795
  • Advanced Physics Homework Help
Replies
2
Views
806
  • Advanced Physics Homework Help
Replies
1
Views
1K
Replies
0
Views
482
  • Advanced Physics Homework Help
Replies
1
Views
967
  • Quantum Interpretations and Foundations
Replies
5
Views
431

Back
Top