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OPTI 511, Spring 2009 Problem Set 7 Prof. R. J. Jones
Due: in class, Thursday, Mar 26
- Electron spinor
An electron is in the spin state:
χ = A
( 3 i 4
)
(a) Determine the normalization constant A.
(b) Find the expectation values of Sx, Sy, and Sz.
NOTE: Although we will probably not discuss some of the following topics in class, you should try to learn the ideas presented here. (You may see some of the most basic elements show up again.) This assignment is also given as an exercise in being able to learn new concepts in Quantum Mechanics on your own. If certain ideas do not make sense, please ask questions in the recitation sections or office hours.
- The Projection Operator
Consider a system with a set of N orthonormal basis states |φn〉. These may be functions of po- sition or momentum, or they may be spinors, or any other complete set of basis states. An arbitrary quantum state in the Hilbert space defined by the basis states can then be expanded as a linear combination of the basis states:
|ψ〉 =
∑^ N n
cn|φn〉,
where the expansion coefficients are in general complex.
As we have previously seen, |ck|^2 is the probability that a measurement of some observable corre- sponding to an operator Qˆ will yield the eigenvalue qk associated with the eigenstate |φk〉 (assuming that |φk〉 is an eigenstate of Qˆ). The expansion coefficient ck is determined by evaluation of the expression ck = 〈φk|ψ〉.
Another way to describe the method to evaluate ck is to define a new operator Pˆk ≡ |φk〉〈φk|. This may look strange, but it is an operator. Consider the N basis states as vectors in N -dimensional Hilbert space. Then, |φk〉〈φk| is the outer product of two vectors, which produces an N × N matrix. As we have seen, operators can be represented by matrices, and matrices represent operators; hence Pˆk ≡ |φk〉〈φk| is indeed an operator.
(a) Evaluate Pˆk|ψ〉 using the quantum states described above, and express your answer in terms of the expansion coefficients and basis states. This operator is called the projection operator because it evaluates the projection (or overlap) of an arbitrary state onto a particular basis state. (Analo- gously, in Cartesian space the inner product R~ · xˆ between an arbitrary vector R~ and the basis vector xˆ determines the projection of R~ onto the ˆx axis.)
(b) Review of matrix manipulations. Assume that two vectors and one matrix are defined as fol- lows:
~u ≡ (a∗^ b∗), ~v ≡
( a b
) , M ≡
( c d e f
) .
Show that the following property holds for these vectors/matrices:
Tr(~u M ~v) = Tr(~v ~u M).
In this expression, “Tr” means trace, a sum over the diagonal elements of a matrix. What you are ex- amining here is the cyclic property of matrix multiplication within a trace: a matrix (or vector) at one end of the string of matrices and vectors within the trace brackets can be moved to the other end with- out affecting the value of the trace. (Note that the trace of a 1 × 1 matrix having just a single element is simply equal to that single matrix element.) This cyclic property holds for larger matrices as well, and for any number of multiplied matrices (but matrix multiplication does have to at least be possible).
(c) Recall that the projection operator Pˆk defined above can be thought of as an N × N matrix. An operator Qˆ associated with some observable in the same Hilbert space is also an N × N matrix. Use the cyclic property of matrix multiplication within a trace to show that Tr( Pˆk Qˆ) is the expectation value of Qˆ for the state |φk〉, i.e., show that
Tr( Pˆk Qˆ) = 〈φk| Qˆ|φk〉 = 〈 Qˆ〉.
(d) Suppose you have two orthonormal basis states |φ 1 〉 and |φ 2 〉 defining a 2D Hilbert space. In the |φ 1 〉, |φ 2 〉 basis, you would write the basis states as the vectors
|φ 1 〉 =
( 1 0
) and |φ 2 〉 =
( 0 1
) .
Evaluate the following expressions by writing each as a 2 × 2 matrix in the |φ 1 〉, |φ 2 〉 basis:
|φ 1 〉〈φ 1 |, |φ 1 〉〈φ 2 |, |φ 2 〉〈φ 1 |, and |φ 2 〉〈φ 2 |.
Note that the othonormality property of the basis states does not apply here in the same way it does for inner products; i.e., each matrix should have at least one non-zero element.
(e) A projection operator can also be defined for an arbitrary quantum state that is a linear su- perposition of eigenstates. Consider the following arbitrary state |ψ〉, a normalized linear sum of the basis states from part (d): |ψ〉 = a|φ 1 〉 + b|φ 2 〉,
where a and b are complex coefficients. Now let the projection operator be defined as Pˆψ = |ψ〉〈ψ|. Evaluate Pˆψ, and write it explicitly as a 2×2 matrix. HINTS: There will not be any non-zero elements. Also, when you write out 〈ψ|, you must remember that a and b are complex, and that (for example) |aφ 1 〉 = a|φ 1 〉 whereas 〈aφ 1 | = a∗〈φ 1 |. Thus, 〈ψ| = a∗〈φ 1 | + b∗〈φ 2 |. The off-diagonal elements of the density matrix will not be identical.
(f ) Evaluate Tr( Pˆψ).
(g) Show that the projection operator Pˆψ acting on an arbitrary state |α〉, has the property that
Introduction to Mixed States and the Density Operator: a short tutorial
The quantum state of an isolated system (for example, an atom alone in empty space) can be charac- terized with a wavefunction. Another way of saying this is that the system is in a “pure state” (not the same thing as a stationary state or energy eigenstate). The time dependence of the system is well- defined via the Schr¨odinger equation, and the quantum state of the particle is describable as a linear sum of energy eigenstates of the system Hamiltonian. We use the wavefunction of the system to gain statistical information about potential measurements, such as expectation values and probabilities of obtaining particular results. All quantum states that we have examined so far in OPTI 511, including the superpositions, have been pure states.
However, in many (actually most) cases a system is not completely isolated from its environment; for example, an atom in a box is really in a “heat bath” of black-body radiation inside the finite- temperature box, or an atom continuously undergoing collisions with many other atoms also can’t be considered isolated. In such cases, the system or particle does not have a wavefunction itself, be- cause the entire wavefunction for the system plus its environment does not factorize into ψS (system coordinates)·ψE (environment coordinates). In such a case, the system of interest is in a so-called “mixed state” rather than a “pure state”. The mixed state can be thought of as a statistical mixture of wavefunctions rather than as a single pure-state wavefunction. Again, a pure state is not the same thing as an energy eigenstate. We have usually worked with coherent linear sums of energy eigen- states: these superpositions are pure states. A mixed state is an incoherent sum, where we work with probabilities rather than probability amplitudes.
Another example of a mixed state is a beam of interacting (colliding) particles. If every particle is in exactly the same quantum state |ψ〉, then we can talk of the beam being in a pure state, |ψ〉. If not — that is, if different particles are in different quantum states, or if collisions randomize the complex expansion coefficients for each particle’s wavefunction — then the beam is in a mixed state.
Mixed states do not have a wavefunction ψ (nor a ket vector |ψ〉), but are described in quantum mechanics by a so-called “density operator” ˆρ. The density matrix is the matrix associated with density operator, and is composed of elements ρnm = 〈φn|ρˆ|φm〉 in some representation defined by basis vectors |φn〉.
As was seen in the preceding problem, a pure state has a definite ket |ψ〉, and thus has an asso- ciated projection operator Pˆψ ≡ |ψ〉〈ψ|. The expectation value of an operator Aˆ in this state is then 〈 Aˆ〉 =Tr( Pˆψ Aˆ).
A mixed state does not have a ket vector, but does have a density operator ˆρ and an associated density matrix. The expectation value for an operator Aˆ for a mixed state described by a density operator ˆρ is 〈 Aˆ〉 =Tr(ˆρ Aˆ). The pure-state projection operator that we worked with above is a special example of a density operator. For either a pure state or a mixed state, the density operator ˆρ for the system can be used to statistically evaluate the probabilities of certain measurement results.
Note that the expectation value of the unit operator ˆ 1 (which is the unity matrix) must always be 1. (As an example, you can evaluate 〈ψ| 1 ˆ|ψ〉 for any normalized quantum state |ψ〉 .) It must therefore be true that Tr(ˆρ)=1 for any density matrix that describes the state of a single particle.
Constructing a density operator and density matrix. The operator ˆρ is Hermitian, thus it will have eigenstates — call these |ξn〉 — with corresponding eigenvalues that we’ll label pn (real numbers). Written in the basis of its eigenstates,
ρˆ =
∑ n
pn|ξn〉〈ξn|.
This form of ˆρ has a simple interpretation: the mixed state can be viewed as a statistical mixture of the |ξn〉 states, with pn being the probability (not probability amplitude) for finding the system in the state |ξn〉. Only in this particular basis will a mixed-state density matrix be diagonal!
Note that 〈 Aˆ〉 = Tr(ˆρ Aˆ) = Tr
(∑ n pn|ξn〉〈ξn|^ Aˆ
)
= Tr
(∑ n pn〈ξn|^ Aˆ|ξn〉
) .
(Remember the cyclic property of matrices within a trace.) The statement that Tr(ˆρ)=1 is then in- terpreted as a statement that probabilities sum to unity: 〈ˆ 1 〉 =
∑ n pn^ = 1, obvious once we associate ˆ 1 with Aˆ.
At this point, it can’t be stressed enough that a mixed state is a mixture of states in a totally different sense than our familiar superpositions of the form “|ψ〉 = ∑ n cn|ξn〉.” This pure state is not a mixture, but a linear combination of basis states. The coefficients cn are complex numbers, thus have a magnitude and a phase. Here, |ψ〉 is a coherent superposition of the |ξn〉 states. In this context, coherent means that definite relative phases are associated with each linear expansion coefficient of the superposition. Such would not be the case if, for example, the system of interest was undergoing collisions or other interactions that would repeatedly randomize the relative phases of the superposi- tion.
Note that when working with pure state wave functions, we are interested in “probability ampli- tudes” such as a and b in the expansion |ψ〉 = a|φ 1 〉 + b|φ 2 〉. These are complex numbers, and must be multiplied with their complex conjugates to give a probability. When working with a density matrix, the diagonal elements are already probabilities, not amplitudes, and are always real, positive numbers that sum to 1.
The concept of a density operator is difficult to understand because it seems so abstract. Yet it is important in that it helps bridge the gap between working with quantum mechanical wavefunctions and everyday incoherent systems that obey familiar statistics and probabilities. The following prob- lems will help you get used to this new concept, and to get a bit more used to working with spin states.
- Density operator for a pure state
In Problem Set 4, problem 1, you determined 〈ˆx〉 for the pure state
|Ψ(t)〉 = (1/
2)|ψ 1 〉e−iE^1 t/¯h^ + (1/
2)|ψ 2 〉e−iE^2 t/¯h,
where |ψ 1 〉 and |ψ 2 〉 were the ground and first excited energy eigenstates (respectively) of the 1- D infinite square well of length L, and E 1 and E 2 were the associated eigen-energies. You found that 〈ˆx〉 oscillates sinusoidally in time back-and-forth in the well. In the present problem, you will solve the same problem using matrix notation and the density operator associated with this pure state.
(a) First, write the state |Ψ(t)〉 in vector notation in a 2-dimensional Hilbert space composed of basis vectors |ψ 1 〉 and |ψ 2 〉. (Neglect all of the other higher energy eigenstates, and just pretend that they don’t exist.) This is obtained by first simply substituting
|ψ 1 〉 =
( 1 0
) and |ψ 2 〉 =
( 0 1
)
into the expression for |Ψ(t)〉, above. Write your answer in terms of E 1 (remember that E 2 = 4E 1 ), and write |Ψ(t)〉 as a single time-dependent column vector.
(b) For a pure state |Ψ(t)〉, the density operator is ˆρ = |Ψ(t)〉〈Ψ(t)|. Using your answer from part (a), write the 2 × 2 density matrix for this quantum state. Don’t forget to take the complex conjugates of the exponentials as needed!
(c) Write the 2 × 2 matrix representation for ˆx in the |ψ 1 〉, |ψ 2 〉 basis. This will be needed in or- der to evaluate 〈ˆx〉 = Tr(ˆρxˆ). Note that the operator ˆx is a time-independent operator, so the matrix elements for ˆx should also be time-independent. There are 4 matrix elements to evaluate:
ˆxab = 〈ψa|xˆ|ψb〉,
where a, b represent the various combinations of the two quantum numbers 1 and 2 and only the time-independent parts of the eigenstates are used (i.e, the matrix elements for ˆx will not have any time-dependent exponential terms). In evaluating these matrix elements, assume that the potential well is centered about the position L/2. Translate Dirac notion to position-space notation (i.e., functions of x), and use the results
∫ (^) L
0
xψ 1 (x)∗ψ 2 (x)dx =
∫ (^) L
0
xψ 2 (x)∗ψ 1 (x)dx =
− 16 L
9 π^2
∫ (^) L
0
x|ψ 1 (x)|^2 dx =
∫ (^) L
0
x|ψ 2 (x)|^2 dx = L/ 2.
Write the full matrix for ˆx in terms of L.
(d) Now evaluate 〈xˆ〉 = Tr(ˆρxˆ). You should obtain an answer that oscillates sinusoidally in time, since there is time dependence in ˆρ. This pure state that is a superposition of two energy eigenstates has oscillatory behavior because of the changing relative phases (in the e−iEt/¯h^ terms of |Ψ(t)〉), and because of interference between the different parts of the superposition.
- Density operator for a mixed state
You’ll now do the same problem as above, but with a mixed state. As an example physical prob- lem, suppose a very large number of atoms are confined to a 1-D potential well. They are all of such low energy that the only two relevant energy eigenstates of the problem are the two lowest energy eigenstates of the potential well (this is not very realistic, but don’t worry about that). Collisions be- tween the particles constitute interactions that are continually randomizing the quantum state of each particle (sudden interactions can suddenly change a quantum state). If we consider an atom in this system, it would be impractical, if not impossible, to assign a quantum state (a ket vector) to it, but we can assign it a density matrix; the density matrix formalism is built to handle such incoherent systems.
(a) Write out the 2 × 2 density matrix for an atom in this system, assuming the probabilities for find- ing the atom in either of the two states |Ψ 1 (t)〉 and |Ψ 2 (t)〉 are equal, where |Ψn(t)〉 = |ψn〉e−iEnt/¯h. Remember the previous definition of the density operator for a mixed state:
ρˆ =
∑ n
pn|ξn〉〈ξn|.
For the problem at hand, n can only be 1 and 2, and the states |ξ〉 are now labeled as |Ψn(t)〉. For our specific problem, ρˆ = p 1 |Ψ 1 (t)〉〈Ψ 1 (t)| + p 2 |Ψ 2 (t)〉〈Ψ 2 (t)|.
When writing out the density matrix, remember to take complex conjugates when needed. Your final answer should be real. (Again, we’re working in the |ψ 1 〉, |ψ 2 〉 basis.)
(b) Using the matrix representation for ˆx found in problem 3(c), evaluate 〈ˆx〉 = Tr(ˆρxˆ) for this mixed state.
(c) Suggest an interpretation for why 〈ˆx〉 is independent of time in this problem, but is dependent on time in problem 3. (HINT: think about the issues of coherence, interference, and environmental coupling.)
- The unit operator
Starting with an expansion of an arbitrary state |ψ〉 as a sum of basis states |φn〉,
|ψ〉 =
∑ n
cn|φn〉,
show that the operator defined by the expression
∑ m |φm〉〈φm|^ can be interpreted as the unit operator ˆ1, with the property that ˆ 1 |ψ〉 = |ψ〉. This expression can come in very useful in evaluating certain
quantities. (See Shoemaker, page 71, posted earlier on the course website.)
