To find the probability of observing state |0⟩ when measuring the quantum state |ψ⟩ = cos(π/4)|0⟩ + e^(iπ)sin(π/4)|1⟩, we need to determine the coefficient associated with the |0⟩ state.
In the given state, the coefficient associated with |0⟩ is cos(π/4). The probability of observing state |0⟩ is equal to the absolute value squared of this coefficient:
P(|0⟩) = |cos(π/4)|^2
Using the trigonometric identity cos^2(θ) = 1/2(1 + cos(2θ)), we can simplify the expression:
P(|0⟩) = |cos(π/4)|^2 = (1 + cos(2(π/4))) / 2
Since cos(π/4) = cos(π/2 - π/4) = cos(π/2 + π/4), we can rewrite the expression as:
P(|0⟩) = (1 + cos(π/2 + π/4)) / 2
Using the values of cos(π/2) = 0 and cos(π/4) = sqrt(2)/2, we can calculate the probability:
P(|0⟩) = (1 + sqrt(2)/2) / 2
Simplifying further:
P(|0⟩) = (2 + sqrt(2)) / 4
Therefore, the probability of observing state |0⟩ when measuring the given quantum state is (2 + sqrt(2)) / 4, which is approximately 0.8536 or 85.36%.