# Solve Quantum Mechanics: Probability of Spin-1/2 Particle

• ProjectionSpin
In summary, the discussion revolved around quantum mechanics and the probability of a spin-1/2 particle. Part 1 focused on the spin-1/2 operator and unit vectors, and how to write normalized eigenvectors in terms of other vectors. In Part 2, the problem involved two unit vectors and the measurement of spin projection in the direction of one vector, leading to a subsequent measurement in the direction of another vector. The solution involved finding the eigenstates and normalizing them, but there was some confusion around the translation of the measurement into mathematical terms.
ProjectionSpin
Quantum Mechanics Help [Probability of a Spin-1/2 Particle]

Part 1) $\hat{S}$ is a spin-1/2 operator, $\vec n$ is a unit vector, $\vert{\psi_\pm}\rangle$ are normalized eigenvectors of $n\cdot\hat{S}$ with eigenvectors $\pm\frac12$. Write $\vert{\psi_\pm}\rangle$ in terms of $\vert{+z}\rangle$ and $\vert{-z}\rangle$.

Part 2) $\vec n_1$ and $\vec n_2$ are unit vectors. A measurement found the projection of spin-1/2 on the direction $\vec n_1$ to be 1/2. Use the results of the previous part to show that a subsequent measurement of the projection of spin on the direction $\vec n_2$ will give 1/2 with probability

$P=\frac12(1+\vec n_1\cdot\vec n_2)$

The attempt at a solution
First I declare what my normal vector $\vec n$ will be, $\vec n = (sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta)$

I then solve for $\sum_i\vec n\cdot\hat{S}_i$ and started looking for the eigenstates, which lead to

$\mu=\pm 1$

Setting this up lead me to
$\langle -z\vert\psi_+\rangle=-e^{i\phi}\frac{\cos\theta-1}{\sin\theta}\langle+z\vert\psi_+\rangle$
which, after normalizing, I then find
$\vert\psi_+\rangle=\cos\frac\theta2\vert+z\rangle+e^{-i\phi}\sin\frac\theta2\vert-z\rangle$

Likewise, when I use $\mu=-1$ I find

$\vert\psi_-\rangle=\sin\frac\theta2\vert+z\rangle-e^{i\phi}\cos\frac\theta2\vert-z\rangle$

**This officially marks the end of part 1**

For part 2, I assumed that "the projection of spin-1/2 on the direction n⃗ 1 to be 1/2" was equivalent to
$\langle \vec n\vert\vec n_1\rangle=\frac12$

But supposably this is not the case, which leaves me to just being confused on where to go.

Last edited:
ProjectionSpin said:
For part 2, I assumed that "the projection of spin-1/2 on the direction n⃗ 1 to be 1/2" was equivalent to
$\langle \vec n\vert\vec n_1\rangle=\frac12$
There is no ##\vec{n}## in part b. The problem mentions the measurement of the projection of the spin in the direction of ##\vec{n}_1##. How do you translate that into mathematical terms?

## What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of particles on a very small scale, such as atoms and subatomic particles. It provides a mathematical framework for understanding the probabilistic nature of these particles and their interactions.

## What is a spin-1/2 particle?

A spin-1/2 particle is a type of subatomic particle, such as an electron or proton, that has an intrinsic angular momentum of 1/2 in units of Planck's constant. This spin property is a fundamental characteristic of particles and plays a crucial role in quantum mechanics.

## How is probability related to quantum mechanics?

In quantum mechanics, the behavior of particles is described by a wave function, which gives the probability of finding a particle in a certain position or state. The square of the wave function represents the probability density, meaning that the higher the value, the more likely it is to find the particle in that state.

## How do you calculate the probability of a spin-1/2 particle?

The probability of a spin-1/2 particle can be calculated using the Pauli spin matrices, which are mathematical operators that describe the spin states of particles. The probability is given by the square of the magnitude of the projection of the particle's spin onto a particular direction or axis.

## What are some applications of quantum mechanics in everyday life?

Quantum mechanics has numerous applications in technology, including semiconductor devices, lasers, and magnetic resonance imaging (MRI). It also plays a crucial role in understanding the behavior of materials at the atomic level and is used in fields such as chemistry, biology, and materials science.

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