Question

Compute in the atomic system of units the normalized states - spin vectors of the particle with a specified projection sy = ​​​​​​​+-1/2 on the y-axis.

Answer 1

Step-by-step explanation:

In the atomic system of units, the normalized states or spin vectors of a particle with a specified projection of ±1/2 on the y-axis can be expressed as:

|+1/2⟩ = (1/√2)[|x⟩ + i|z⟩]

|-1/2⟩ = (1/√2)[|x⟩ - i|z⟩]

Here, |x⟩, |y⟩, and |z⟩ are the basis vectors for the x, y, and z axes, respectively. The factor of 1/√2 ensures that the spin vectors are normalized.

Note that these spin vectors are complex linear combinations of the basis vectors, indicating that the particle has non-zero probability amplitudes for being in both the x and z spin states, in addition to the specified y spin state.

Answer 2

Your welcome

Step-by-step explanation:

In the atomic system of units, the normalized states - spin vectors of a particle with a specified projection sy = ±1/2 on the y-axis can be calculated using the following steps:

1. Start with the spin state vector notation: |sy⟩.

2. Normalize the state vector by dividing it by the square root of 2: |sy⟩ / √2.

3. Since the projection can be either +1/2 or -1/2, we can write the normalized states as:

For sy = +1/2: |+1/2⟩ = |sy⟩ / √2

For sy = -1/2: |-1/2⟩ = |sy⟩ / √2

4. Simplify the expression by substituting the value of |sy⟩ with the y-component basis vector: |+1/2⟩ = |y+⟩ / √2 and |-1/2⟩ = |y-⟩ / √2.

So, in the atomic system of units, the normalized states - spin vectors with a specified projection sy = ±1/2 on the y-axis are:

For sy = +1/2: |+1/2⟩ = |y+⟩ / √2

For sy = -1/2: |-1/2⟩ = |y-⟩ / √2

These vectors represent the possible spin states of a particle along the y-axis, with the specified projections. The normalization ensures that the total probability of finding the particle in any spin state is 1.