PH425 Spins Homework 5 Due 4 pm. particles is prepared in the state: + + i 3 13

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1 PH45 Spins Homework 5 Due 4 pm REQUIRED: 1. A beam of spin- 1 particles is prepared in the state: ψ + + i 1 1 (a) What are the possible results of a measurement of the spin component S z, and with what probabilities would they occur? Check Beasts: Check that you have the right beast. The (normalized) state we are given is ψ i 1, where { +, } is the spin-1/ z-basis. The possible values of the measurement of the z-component of angular momentum, i.e. ±( h/). You just have to know this. There is nothing to calculate. The probabilities to get them in a measurement are given by the squares of the norms of the coefficients in the expansion (of the particular state) in this basis. Since the state itself is already given in that basis, the probabilities are simply the squares of the norms of the coefficients, ie. P + + ψ and P ψ i These coefficients are positive, real scalars that add up to 1, as they ought to. (Since these are the only possible values/states, we must find one of them in a measurement.) (b) What are the possible results of a measurement of the spin component S x, and with what probabilities would they occur? Check Beasts: Check that you have the right beast. The possible values that can be measured are again ± h. Getting them in a measurement means ending up in the respective eigenstate, + x or x. The probability of this happening, for the given state ψ, can be obtained as P +x x + ψ, and P x x ψ. This can also be stated as follows. If the state ψ is written in terms of Ŝx, which is { + x, x }, the probabilities are found from the expansion coefficients (by squaring their absolute values). The coefficients are precisely x ± ψ. 1

2 The + x coefficient is: x + ψ 1 x i 1 x i 1. This is the + x coefficient in the expansion of ψ in the ± x basis. The probability of a measuring the S x spin component to be h is the norm squared of this coefficient. P(S x h ( ) i ) ( + i ) The P(S x h ) is calculated the same way (or you can subtract P(S x h ) from 1) and is 1/. (c) Use Another Representation: Plot histograms of the predicted measurement results from parts (a) and (b). For part (a): For part (b):. Consider the three quantum states: ψ i 5 ψ i 5 ψ i 5

3 (a) For each of the ψ i above, calculate the probabilities of spin component measurements along the x, y, and z-axes. First, you should check that the states are normalized. You can do this by taking the braket of each ψ with itself to see if you get 1. These states are normalized. The probability to end up in an state a i is given by a i ψ i. The states ψ i are given in the Ŝz basis, so probabilities for measurements of the z component of spin are practically read off: For ψ 1 : P(S z h ( ) 4 ) + ψ i 16/5, 5 P(S z h ( ) ψ i i ) ( i ) 9/ Other probabilities are a little more interesting, and here we deal in detail with those related ( to the measurements ) of Ŝy. Following the general prescription, with ± y 1 + ± i, (don t forget to complex conjugate the coefficients when taking the bra!) y + ψ 1 1 ) ( 4 ( + i i ) 5 y ψ 1 1 ) ( 4 ( + + i i ) ( i ) ( i ( i ) ) 5 ) ( i 5 7 5, 1 5. The probabilities are the squares of the norms of these coefficients, P(S y h ) 49/50 and P(S y h ) 1/50 (adding up to 1). For ψ, the sign of i in the second parenthesis above is changed to minus, and we get y + ψ 4 ( ) i i , and y ψ 7 5. Note that the coefficients, and thus probabilities, are swapped compared to the previous ones; now P(S x h ) is 1/50, and P(S x h ) is 49/50. For ψ : y + ψ 1 ) ( + i ( i 5 ) 4 5 ) ( i 5 ) y ψ 1 ( + + i ( ) i i 5 1 5, ( ) i i

4 The probabilities are the same as those when the system is measured in the state ψ, above. The calculations for Ŝx are carried out exactly the same way. (b) Look For a Pattern (and Generalize): Use your results from (a) to comment on the importance of the overall phase and of the relative phases of the quantum state vector. Notice that the ψ ψ e iπ ψ, i.e. these states differ only by an overall phase. As expected, these states are physically the same and therefore all of the probabilities that we calculated for these two states are the same. Now let us observe the relative phases between the components of these states. With phases of components written out explicitly, ψ 1 e i e i π (since e i π i, e i 0 1), while for the other two states we have ψ e i e i π, ψ e i π + + e i π e i π (e i e i π ). The difference between phases of the components, the relative phase, for ψ 1 is π, while for ψ (and ψ ) it is π. States that have different relative phases, like ψ 1 vs. ψ (or vs. ψ ) have different probabilities. These states are physically inequivalent. Note, however, that since the coefficients in the z-basis have the same norms squared, the probabilities for spin up and spin down in the z-orientation are the same for all three states. It is not until we calculate probabilities in the x- or y-orientations that we see the inequivalence of the first state from the other two.. With the Spins simulation set for a spin 1/ system, measure the probabilities of all the possible spin components for each of the unknown initial states ψ i (i 1,,, 4). (a) Use your measured probabilities to find each of the unknown states as a linear superposition of the S z -basis states + and. You have experimentally obtained the relative probabilities with for each possible value of the spin components for the unknown initial states ψ i (i 1,,, 4). To find these initial states, expressed in the z-basis, { +, }, we use a general form for a quantum state, ψ i a i + + b i e iγ i. where a i and b i are real and positive unknowns and γ i is an unknown angle. (Notice that you can use the freedom in the overall phase for the state to set the phase of the coefficient of + to be zero.) 4

5 Here, I will show how to find ψ 4 and then make some general comments about the easier cases. The experimental results for the probabilities are, P(S z h/) + ψ P(S z h/) ψ 4 4 P(S x h/) x + ψ 4 1 P(S x h/) x ψ 4 1 P(S y h/) y + ψ P(S y h/) y ψ Since ψ 4 is given in the z-basis, use the s z probabilities first. + ψ 4 ( ) + a b 4 e iγ a 4 (a 4 ) 1 4, ( ) ψ 4 a b 4 e iγ 4 4 b 4 e iγ 4 (B 4 ) 4. So we have found that a 4 1 and b 4 about γ 4. but we did not get any information Now I will use the probabilities from the S x and S y components. Recall that I am working in the z-basis, so I use expressions for ± x and ± y in that basis. x + ψ 4 1 )( 1 ) ( eiγ eiγ eiγ 4 ( ) ( ) e iγ 4 + eiγ 4 5

6 1 (1 + + ( e iγ 4 + e ) ) iγ ( + ) cos γ ( + ) cos γ , cos γ 4 0. This yields two possible solutions: γ 4 ± π. To distinguish between these two possible angles, I need to calculate another probability. But using x will give me the same result as above. Why do you think that is?) Instead, I can use one of the the probabilities of the S y components. y + ψ 4 1 )( 1 ) ( + + i + + eiγ i eiγ 4 ( ) ( ) 1 i 1 e iγ 4 + i eiγ 4 1 ( 4 + i ( )) e iγ 4 e iγ ( ) sin γ , sin γ 4 1. Therefore, I conclude that γ 4 π This agrees with the previous calculation. Thus we have found that the unknown state is: ψ ei π or ψ i To find ψ one runs through much the same procedure. ψ 1 + y ψ ψ π e i

7 1 + + i (b) Articulate a Process: Write a set of general instructions that would allow another student in next year s class to find an unknown state from measured probabilities. i. Represent the unknown state with its phase factor, as ψ a + + be iγ. ii. Calculate the unknowns a and b by projecting onto the z-basis. + ψ ψ a b iii. Compare these equations to the probabilities from the experiment (which are the squares of the norms of these equations) and thereby identify a and b. iv. Project the unknown state ψ onto the the bra + x and compare to the probabilities from the experiment. This will give one equation for the unknown angle γ. v. If there is still an ambiguity that says γ is one of two possible angles, repeat the step above by projecting onto the bra + y, which will give a second equation for the unknown angle γ which will remove the ambiguity. (c) Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results. From the spin reference sheet, we see that the spin-1/ state which is pure spin up according to a Stern-Gerlach device oriented in the ˆn direction is given in the z-basis by: + n cos θ + + sin θ eiφ where θ and φ are the angles that describe the orientation of the Stern-Gerlach device in spherical coordinates. By comparing this expression to our expression for the unknown state, I infer the direction for which the unknown state is spin up. By running the unknown state through a Stern-Gerlach device in this orientation, we should get all of the particles coming out the up port with 100% probability. (d) Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not? 7

8 4. Consider a quantum system described by an orthonormal basis a 1, a, and a. The system is initially in a state: ψ in i a 1 + a Find the probability that the system is measured to be in the final state: ψ out 1 + i a a a Check Beasts: Check that you have the right beast. The probability for the transition from a state ψ in to a state ψ out is given by ψ out ψ in. This is also the probability to measure ψ out on a system that is initially in ψ in. Since our states are given in the same basis, calculating this is a matter of finding absolute value squared of the product of the corresponding coefficients. ψ out ψ in ( 1 i a a + 1 ) ( ) i a a a 1 + i + 18 Then the probability is, using 18 9, P ψ out ψ in 1 + i + + i ( ) ( ) + i i 5 9. Beasts: This probability is a real, positive scalar between 0 and 1, which is appropriate for a probability. 8