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The problem asks us to find the probability that the spin component will be "up" along the Y-direction for the given quantum state \( |\psi_3\rangle \). 

### Step-by-Step Solution

#### 1. Quantum State of \( |\psi_3\rangle \)
We are given the state \( |\psi_3\rangle \) as:

\[
|\psi_3\rangle = |+\rangle - 2 e^{-i\frac{\pi}{4}} |-\rangle
\]

This is a superposition of the eigenstates \( |+\rangle \) and \( |-\rangle \), which are typically associated with the spin component along the Z-direction.

#### 2. Transformation to Y-basis
To find the probability for the spin component to be "up" along the Y-direction, we need to express \( |\psi_3\rangle \) in terms of the spin-up and spin-down states along the Y-axis, denoted \( |+\rangle_y \) and \( |-\rangle_y \).

The Y-basis states can be written in terms of the Z-basis states \( |+\rangle \) and \( |-\rangle \) as:

\[
|+\rangle_y = \frac{1}{\sqrt{2}} (|+\rangle + i |-\rangle)
\]
\[
|-\rangle_y = \frac{1}{\sqrt{2}} (|+\rangle - i |-\rangle)
\]

#### 3. Expanding \( |\psi_3\rangle \) in Terms of \( |+\rangle_y \) and \( |-\rangle_y \)
Now, let's express \( |\psi_3\rangle \) in the Y-basis.

\[
|\psi_3\rangle = |+\rangle - 2 e^{-i\frac{\pi}{4}} |-\rangle
\]

Substitute the Z-basis states in terms of the Y-basis states:

\[
|\psi_3\rangle = \left[ \frac{1}{\sqrt{2}} (|+\rangle_y + |-\rangle_y) \right] - 2 e^{-i\frac{\pi}{4}} \left[ \frac{1}{\sqrt{2}} (|+\rangle_y - i |-\rangle_y) \right]
\]

Simplify the expression by distributing and combining like terms:

\[
|\psi_3\rangle = \frac{1}{\sqrt{2}} |+\rangle_y + \frac{1}{\sqrt{2}} |-\rangle_y - \frac{2 e^{-i\frac{\pi}{4}}}{\sqrt{2}} |+\rangle_y + \frac{2 i e^{-i\frac{\pi}{4}}}{\sqrt{2}} |-\rangle_y
\]

Now combine the terms with \( |+\rangle_y \) and \( |-\rangle_y \):

\[
|\psi_3\rangle = \frac{1 - 2 e^{-i\frac{\pi}{4}}}{\sqrt{2}} |+\rangle_y + \frac{1 + 2 i e^{-i\frac{\pi}{4}}}{\sqrt{2}} |-\rangle_y
\]

#### 4. Probability for Spin "Up" Along Y-axis
The probability for the spin to be "up" along the Y-axis is the square of the amplitude of the \( |+\rangle_y \) term. The amplitude of \( |+\rangle_y \) is:

\[
\frac{1 - 2 e^{-i\frac{\pi}{4}}}{\sqrt{2}}
\]

Now, compute the probability:

\[
P_{\text{up}, y} = \left| \frac{1 - 2 e^{-i\frac{\pi}{4}}}{\sqrt{2}} \right|^2
\]

First, calculate \( 1 - 2 e^{-i\frac{\pi}{4}} \):

\[
e^{-i\frac{\pi}{4}} = \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}
\]

\[
1 - 2 \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) = 1 - \sqrt{2} + i \sqrt{2}
\]

Now, calculate the magnitude squared of this expression:

\[
\left| 1 - \sqrt{2} + i \sqrt{2} \right|^2 = (1 - \sqrt{2})^2 + (\sqrt{2})^2
\]

\[
= (1 - \sqrt{2})^2 + 2 = (1 - 2\sqrt{2} + 2) + 2 = 5 - 2\sqrt{2}
\]

Finally, the probability is:

\[
P_{\text{up}, y} = \frac{5 - 2\sqrt{2}}{2}
\]

Thus, the probability for the spin component to be "up" along the Y-direction is:

\[
P_{\text{up}, y} = \frac{5 - 2\sqrt{2}}{2}
\]

### Do you need further details or clarifications?

### Relative Questions:
1. What is the significance of the Y-axis in quantum spin measurements?
2. How can one interpret complex coefficients in quantum superposition?
3. Why are basis transformations important in quantum mechanics?
4. What role do Z-basis states play when measuring along different axes?
5. How do probabilities arise from amplitudes in quantum mechanics?

### Tip:
In quantum mechanics, always check that the total probability sums to 1 when calculating probabilities in a given basis.

For JUST state |ψ3⟩, find the probability that the spin component will be 'up' along the Y-direction. Use bra-ket notation in your calculations.

Solve a quantum mechanics problem to find the probability that the spin component will be 'up' along the Y-axis for the quantum state |ψ3⟩. Using bra-ket notation and complex number arithmetic, this problem is suitable for advanced undergraduate students studying quantum mechanics.

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