Total Angular Momentum Of ³He/³H: A Physics Discussion

Oct 28, 2025 by Admin 55 views

Hey physics enthusiasts! Let's dive into an interesting problem involving nuclear physics and angular momentum. We're going to explore how the total angular momentum is determined when we form Helium-3 (³He) or Tritium (³H) by adding a nucleon (either a proton or a neutron) to Deuterium (²H). This might sound a bit complex, but we'll break it down step by step. So, buckle up and let's get started!

Understanding the Basics

Before we jump into the specifics, let's clarify some key concepts. First, we need to understand what angular momentum is. In simple terms, it’s a measure of how much an object is rotating. In quantum mechanics, angular momentum is quantized, meaning it can only take on specific discrete values. These values are typically represented by quantum numbers.

Next, let's talk about Deuterium (²H). Deuterium is an isotope of hydrogen, consisting of one proton and one neutron in its nucleus. The problem states that Deuterium has a spin of 1 and even parity. Spin is another form of angular momentum, intrinsic to the particle itself, and parity is related to how the wavefunction of the nucleus behaves under spatial inversion (flipping the coordinates). Even parity means the wavefunction remains unchanged when the coordinates are flipped.

We're adding either a proton or a neutron to Deuterium to form Helium-3 (²He) or Tritium (³H), respectively. The added nucleon has an orbital angular momentum (denoted by ℓ) relative to the center of mass of the Deuterium nucleus. This orbital angular momentum is also quantized and contributes to the total angular momentum of the resulting nucleus.

Now, the big question: How do we figure out the possible values of the total angular momentum? This involves combining the angular momenta of the Deuterium nucleus and the added nucleon. This is where the rules of angular momentum coupling in quantum mechanics come into play. The heart of the problem lies in understanding how these different angular momenta interact and combine to give us the final possible values. We’ll need to consider both the spin of the particles and their orbital angular momentum, ensuring we follow the quantum mechanical rules for addition.

Determining Possible Values

To figure out the possible values of the total angular momentum, we need to consider the angular momenta involved and how they combine. We have the spin of Deuterium (²H), the spin of the added nucleon (either a proton or a neutron), and the orbital angular momentum (ℓ) of the added nucleon relative to the ²H nucleus. Let's break this down:

Deuterium Spin: Deuterium has a spin of 1. This means its spin quantum number, s_D, is 1.
Added Nucleon Spin: Both protons and neutrons have a spin of 1/2. So, the spin quantum number for the added nucleon, s_N, is 1/2.
Orbital Angular Momentum (ℓ): This is the angular momentum due to the motion of the added nucleon around the Deuterium nucleus. ℓ can take on integer values: 0, 1, 2, and so on. Each value corresponds to a different spatial distribution of the nucleon around the nucleus.

Now, we need to combine these angular momenta. First, we combine the spin of Deuterium and the spin of the added nucleon. This gives us a possible range of total spin values. The rule for combining two angular momenta is that the total angular momentum quantum number, S, can range from the absolute difference of the individual quantum numbers to their sum, in integer steps. Mathematically:

|s_D - s_N| ≤ S ≤ s_D + s_N

Plugging in our values, we get:

|1 - 1/2| ≤ S ≤ 1 + 1/2

1/2 ≤ S ≤ 3/2

So, the possible values for the total spin quantum number S are 1/2 and 3/2. This means the combined spin can be either 1/2 or 3/2.

Next, we need to combine this total spin S with the orbital angular momentum ℓ. Again, we use the same rule for combining angular momenta. The total angular momentum quantum number, J, can range from the absolute difference of S and ℓ to their sum, in integer steps:

|S - ℓ| ≤ J ≤ S + ℓ

This is where things get interesting because the possible values of J depend on the value of ℓ. We need to consider different values of ℓ separately.

Possible J Values for Different ℓ

Let's explore the possible values of the total angular momentum J for different values of the orbital angular momentum ℓ. Remember, we have two possible values for the combined spin S: 1/2 and 3/2.

Case 1: ℓ = 0

When ℓ = 0, there is no orbital angular momentum. This means the added nucleon is essentially in a spherically symmetric orbit around the Deuterium nucleus. We now combine our two possible S values with ℓ = 0:

For S = 1/2:

|1/2 - 0| ≤ J ≤ 1/2 + 0

1/2 ≤ J ≤ 1/2

So, J = 1/2.
For S = 3/2:

|3/2 - 0| ≤ J ≤ 3/2 + 0

3/2 ≤ J ≤ 3/2

So, J = 3/2.

Thus, when ℓ = 0, the possible values for the total angular momentum J are 1/2 and 3/2.

Case 2: ℓ = 1

When ℓ = 1, the added nucleon has one unit of orbital angular momentum. This corresponds to a more complex spatial distribution around the Deuterium nucleus. Again, we combine our two possible S values with ℓ = 1:

For S = 1/2:

|1/2 - 1| ≤ J ≤ 1/2 + 1

1/2 ≤ J ≤ 3/2

So, J can be 1/2 or 3/2.
For S = 3/2:

|3/2 - 1| ≤ J ≤ 3/2 + 1

1/2 ≤ J ≤ 5/2

So, J can be 1/2, 3/2, or 5/2.

Therefore, when ℓ = 1, the possible values for the total angular momentum J are 1/2, 3/2, and 5/2.

Case 3: ℓ = 2

Let's push this further and consider ℓ = 2. This corresponds to an even more complex orbital motion of the added nucleon. Combining our S values with ℓ = 2:

For S = 1/2:

|1/2 - 2| ≤ J ≤ 1/2 + 2

3/2 ≤ J ≤ 5/2

So, J can be 3/2 or 5/2.
For S = 3/2:

|3/2 - 2| ≤ J ≤ 3/2 + 2

1/2 ≤ J ≤ 7/2

So, J can be 1/2, 3/2, 5/2, or 7/2.

Thus, when ℓ = 2, the possible values for the total angular momentum J are 1/2, 3/2, 5/2, and 7/2.

Parity Considerations

We also need to consider parity. The total parity of the resulting nucleus is the product of the parities of the individual components. Deuterium has even parity (+1), and the added nucleon also has an intrinsic parity of +1. However, the orbital motion introduces a parity factor of (-1)^ℓ. So, the total parity is:

Parity_Total = Parity_Deuterium × Parity_Nucleon × (-1)^ℓ

Parity_Total = (+1) × (+1) × (-1)^ℓ = (-1)^ℓ

This means that for even values of ℓ (0, 2, etc.), the total parity is even (+1), and for odd values of ℓ (1, 3, etc.), the total parity is odd (-1). This parity consideration further constrains the possible states of the ³He or ³H nucleus.

Summary of Possible J Values

Let's summarize our findings. The possible values of the total angular momentum J, along with the corresponding parity, are:

ℓ = 0: J = 1/2, 3/2; Parity = +1
ℓ = 1: J = 1/2, 3/2, 5/2; Parity = -1
ℓ = 2: J = 1/2, 3/2, 5/2, 7/2; Parity = +1

And so on. As ℓ increases, the number of possible J values also increases.

Conclusion

Calculating the possible values of the total angular momentum when forming ³He or ³H involves understanding angular momentum coupling in quantum mechanics. We considered the spin of Deuterium, the spin of the added nucleon, and the orbital angular momentum of the added nucleon. By combining these angular momenta according to the rules of quantum mechanics, we determined the possible values for the total angular momentum J. We also considered parity, which further constrains the possible states. This analysis provides valuable insights into the structure and properties of light nuclei.

I hope this discussion helped you understand how to approach such problems. Nuclear physics can be fascinating, and understanding angular momentum is a crucial step in unraveling the complexities of the nucleus. Keep exploring, guys, and happy learning!