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Spring 2017 115C Group 1

115C: Quantum Group 1 Commutators, Vector Operators, and Measurement

Jared Pagett April 26, 2017

1 and Einstein Notation

As you discovered on the last homework, writing out huge lists of components can make commutators tedious – especially when dealing with cross products. You can use Einstein notation to make life easier on yourself. For an introduction to index notation and Einstein notation, see the following: This may be obvious to some of you, but in case you haven’t seen it, a vector v represented in the {eˆi} (which, for convenience, we will take as orthonormal) with components {vi} can be written X v = vieˆi i Note that for two vectors A and B, we can write X X X X A · B = (Aieˆi) · (Bjeˆj) = Ai(eˆi · eˆj)Bj = AiδijBj = AiBi ij ij ij i where we sum over i = x, y, z. In Einstein notation we suppress the sign, and take on the convention that if an index is repeated there is an implied sum over it. For example, we would write for the dot above A · B ≡ AiBi In case you aren’t familiar with cross products in summation notation, note that the of vectors A and B takes the form A × B = ijkAjBkeˆi where ijk is the Levi-Civita symbol. This symbol is antisymmetric under the exchange of any two of its indices, meaning that ijk = −ikj = kij, and so on. th th Finally, some side notes about matrices. We write the entry of a A in the i row, j column as Aij (or aij). The (Tr) of a matrix is the sum of its diagonal components, and so can be written

Tr[A] = Aijδij = Aii The product of two matrices A and B is generally written

AB = (AB)ik = AijBjk

2 Index Notation Practice

These identities might prove useful:

ijkimn = δjmδkn − δjnδkm & jmnimn = 2δij [AB, C] = A [B,C] + [A, C, B] [A, BC] = B [A, C] + [A, B] C For each of the following, try not to write out individual components – the idea here is to practice using index notation.

1 Spring 2017 Physics 115C Group 1

2.1 Proving the Bread & Butter

Given that L = r × p, and knowing that [ri, rj] = 0, [pi, pj] = 0, and [ri, pj] = i~δij, show that

[Li,Lj] = i~ijkLk This may not be the best problem to illustrate that index notation makes life easy – still, practice is good. Next, verify that [Li, xj] = i~ijkxk and [Li, pj] = i~ijkpk

2.2 Again! Remember that lovely problem from your homework where you computed [L · S, L]? If you found yourself writing out components, try it again with Einstein notation. Hopefully you’ll find it much easier this time. You may use the fact that [Li,Sj] = 0 and that [Li,Lj] = i~ijkLk.

2.3 Quirks of Vector Operators On the last homework set, you dealt a lot with L and S – vectors which have operators as their components. Sometimes this can introduce quirks that you’re not used to dealing with – if you’re not careful, the simplest vector identities can take on a far more sinister nature. To do the following, make use of the fact that you can switch the order of any two operators at the cost of those terms’ commutator:

aibj = bjai + [ai, bj]

1. For general vector operators a and b, (a) How is a · b related to b · a? (b) How is a × b related to b × a? Be careful not to assume anything about the way components of a and b behave! 2. Compute the following: (a) L × L. It’s not what you think! (b) L · (L × S). Again – it’s not what you think! You may use the result of the previous part. 3. Convince yourself that [A · B, C] 6= A · [B, C] + [A, C] · B. If you don’t see it, write out one of the commutators on the right side and ask yourself if what you’ve written makes sense.

2.4 More Practice Show the following using Einstein notation. 1. Ah, your good friends the . Given that [σi, σj] = 2iijkσk and that {σi, σj} = σiσj + σjσi = 2δijI, (a) Show that 1 σ σ = ({σ , σ } + [σ , σ ]) i j 2 i j i j (b) With the given information and the result of the previous part, show that

(σ · a) · (σ · b) = (a · b)I + iσ · (a × b)

for σ = (σ1, σ2, σ3) and general 3-vectors a and b. (c) Use the results of the previous part to determine the result of (σ · nˆ)2 for general unit vector nˆ.

2 Spring 2017 Physics 115C Group 1

2. Show that the triple product is under cyclic permutation – that is, that for vectors a, b, and c with real coefficients, a · (b × c) = c · (a × b)

3. Show that for n × n matrices Ω, Γ, and Λ, (a) Tr[ΩΓ] = Tr[ΓΩ] (b) Tr[ΩΓΛ] = Tr[ΓΛΩ] = Tr[ΛΩΓ] – i.e., that the trace is invariant under cyclic permutation.

4. Without using the result of the last part, show that the trace is invariant under unitary transformations – that for general matrix Ω and unitary U, Tr[U †ΩU] = Tr[Ω]. Recall that a matrix U is unitary if and only if U †U = I.

3 Measurement

Something really important to quantum mechanics is the concept of measurement. This section is intended to be review, but hopefully it proves helpful. First, some clarification. The result of a measurement is an eigenvalue. Yes, the measurement causes your wavefunction to collapse to an eigenstate of the measured observable, but the thing you get back from your measurement is the eigenvalue corresponding to the eigenstate it collapsed to. Let’s say we start with a state |Ψi and make a measurement of an observable Aˆ. Suppose that our state collapses to eigenstate |ai of Aˆ such that Aˆ|ai = a|ai. • The result of our measurement will be a. • The probability of having obtained this result from a measurement of Aˆ on |Ψi is given by

P (a) = |ha|Ψi|2

Unless it’s incredibly convenient (say, if you can use a Clebsch-Gordon table or something), there’s no need to go rewrite |Ψi in the basis of eigenstates of Aˆ and pick out the coefficient of the |ai component to square and obtain the probability. Don’t do it! You can simply compute the inner product right off the bat. Some of you have a fixation on the coefficients of the expansion and do unimaginable amounts of algebra to obtain them all – just jump right to the inner products. • The system is now in state |ai. Further measurements of Aˆ will give as a result the eigenvalue a, should we make those measurements without allowing our state to evolve in time. Whether this all seems obvious or intimidating, practice – make sure it sticks.

Practice Measurement: Given the χ0 below represented terms of the z basis states χ+ and χ−, what is the probability of obtaining −~/2 for a measurement of the component of spin in the x-direction?

χ0 = A [(1 + i)χ+ + (3 − i)χ−]

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