Question

A proton is in the spin state α y . What is the probability that measurement finds each of the following? (a) S y =−ℏ/2 (b) S x =+ℏ/2 (c) S x =−ℏ/2 (d) S z =+ℏ/2 (e) S z =−ℏ/2 Answers (a) 0 (b) 2 1 (c) 2 1 (d) 2 1 (e) 2 1

Answer 1

Step-by-step explanation:

Given that the proton is in the spin state αy, we can express it as:

|αy⟩ = 1/√2 * (|↑⟩ + i|↓⟩)

where |↑⟩ and |↓⟩ represent the spin-up and spin-down states along the y-axis, respectively.

Now, let's calculate the probabilities for each measurement:

(a) S_y = -ℏ/2:

To find the probability of measuring Sy = -ℏ/2, we need to determine the projection of the state αy onto the eigenstate |αy⟩ = |↓⟩ along the y-axis.

P(Sy = -ℏ/2) = |⟨↓|αy⟩|^2

Substituting the values, we get:

P(Sy = -ℏ/2) = |⟨↓|αy⟩|^2 = |⟨↓|(1/√2 * (|↑⟩ + i|↓⟩))|^2

= (1/√2) * |i|^2

= 0

Therefore, the probability of finding Sy = -ℏ/2 is 0.

(b) S_x = +ℏ/2:

To find the probability of measuring Sx = +ℏ/2, we need to determine the projection of the state αy onto the eigenstate |αx⟩ = (1/√2) * (|↑⟩ + |↓⟩) along the x-axis.

P(Sx = +ℏ/2) = |⟨αx|αy⟩|^2

Substituting the values, we get:

P(Sx = +ℏ/2) = |⟨αx|αy⟩|^2 = |⟨αx|(1/√2 * (|↑⟩ + i|↓⟩))|^2

= (1/2) * |1|^2

= 1/2

Therefore, the probability of finding Sx = +ℏ/2 is 1/2.

(c) S_x = -ℏ/2:

To find the probability of measuring Sx = -ℏ/2, we need to determine the projection of the state αy onto the eigenstate |αx⟩ = (1/√2) * (|↑⟩ + |↓⟩) along the x-axis.

P(Sx = -ℏ/2) = |⟨αx|αy⟩|^2

Substituting the values, we get:

P(Sx = -ℏ/2) = |⟨αx|αy⟩|^2 = |⟨αx|(1/√2 * (|↑⟩ + i|↓⟩))|^2

= (1/2) * |1|^2

= 1/2

Therefore, the probability of finding Sx = -ℏ/2 is 1/2.

(d) S_z = +ℏ/2:

To find the probability of measuring Sz = +ℏ/2, we need to determine the projection of the state αy onto the eigenstate |αz⟩ = (1/√2) * (|↑⟩ + e^(iπ/2)|↓⟩) along the z-axis.

P(Sz = +ℏ/2) = |⟨αz|αy⟩|^2

Substituting the values, we get:

P(Sz = +ℏ/2) = |⟨αz|αy⟩|^2 = |⟨αz|(1/√2 * (|↑⟩ + i|↓⟩))|^2

= (1/2) * |1|^2

= 1/2

Therefore, the probability of finding Sz = +ℏ/2 is 1/2.

(e) S_z = -ℏ/2:

To find the probability of measuring Sz = -ℏ/2, we need to determine the projection of the state αy onto the eigenstate |αz⟩ = (1/√2) * (|↑⟩ + e^(-iπ/2)|↓⟩) along the z-axis.

P(Sz = -ℏ/2) = |⟨αz|αy⟩|^2

Substituting the values, we get:

P(Sz = -ℏ/2) = |⟨αz|αy⟩|^2 = |⟨αz|(1/√2 * (|↑⟩ + i|↓⟩))|^2

= (1/2) * |1|^2

= 1/2

Therefore, the probability of finding Sz = -ℏ/2 is 1/2.

To summarize:

(a) P(Sy = -ℏ/2) = 0

(b) P(Sx = +ℏ/2) = 1/2

(c) P(Sx = -ℏ/2) = 1/2

(d) P(Sz = +ℏ/2) = 1/2

(e) P(Sz = -ℏ/2) = 1/2