et |ϕ⟩=a|0⟩+b|1⟩ where a and b are nonnegative real numbers. We know that if we measure |ϕ⟩ in the standard basis, the probability of getting a 0 is 925. What is |ϕ⟩?
To determine the values of a and b in the state vector |ϕ⟩, we can use the fact that the coefficients must satisfy certain conditions. In this case, we are given that the probability of measuring |ϕ⟩ in the standard basis and obtaining a 0 is 925.
The probability of measuring a particular outcome is equal to the square of the absolute value of the coefficient of the corresponding state in the superposition. In other words, for the standard basis states |0⟩ and |1⟩, the probabilities are given by:
P(0) = |a|^2
P(1) = |b|^2
Since the coefficients must be nonnegative, we have:
|a|^2 = 925
To solve for a, we take the square root of both sides:
|a| = √925
Since a is a nonnegative real number, we can simplify this to:
a = √925
Similarly, we have:
|b|^2 = 1 - |a|^2
|b|^2 = 1 - 925
|b|^2 = -924
Since b is also a nonnegative real number, we conclude that b must be 0. Therefore, the state vector |ϕ⟩ can be written as:
|ϕ⟩ = √925|0⟩ + 0|1⟩
|ϕ⟩ = √925|0⟩