Quantum Mechanics 101 · Unit 3: The Mathematical Machinery

Quantum Academy · Chapter 14: Operators, Observables & Dirac Notation

The grammar of quantum mechanics: kets, eigenvalues and commutators

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Instructions. Work through every question with the lesson closed, then check yourself. Use ħ = 1.0546×10⁻³⁴ J·s, h = 6.626×10⁻³⁴ J·s, mₑ = 9.109×10⁻³¹ kg and 1 eV = 1.602×10⁻¹⁹ J. Write every matrix row by row: the matrix with rows (a, b) and (c, d). Recall σx has rows (0, 1) and (1, 0), σy has rows (0, −i) and (i, 0), and σz has rows (1, 0) and (0, −1). Show every step in the calculation questions. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Ket
2.Bra
3.Inner product
4.Hilbert space
5.Observable
6.Hermitian operator
7.Eigenvalue
8.Eigenstate
9.Eigenbasis
10.Expectation value
11.Pauli matrices
12.Commutator

A.The partner ⟨ψ| of a ket: the same amplitudes laid out as a row and complex-conjugated, used to form inner products.
B.Dirac's symbol |ψ⟩ for a quantum state: a vector in Hilbert space, written for a finite system as a column of complex amplitudes.
C.The average result ⟨ψ|Â|ψ⟩ of measuring an observable on many identically prepared systems.
D.The bracket ⟨φ|ψ⟩, a complex number measuring the overlap of two states; its squared magnitude is a probability.
E.A number a for which Âψ = aψ has a non-zero solution ψ; for an observable, one of the possible results of a measurement.
F.A state that an operator only multiplies by a number; for an observable, a state in which that quantity has a definite value.
G.A measurable physical quantity, such as energy, momentum or spin, represented in quantum mechanics by a Hermitian operator.
H.[Â, B̂] = ÂB̂ − B̂Â, which measures whether the order of two operators matters; zero means the observables are compatible.
I.The three Hermitian two-by-two matrices σx, σy and σz, each with eigenvalues ±1, describing spin one half and qubits.
J.A complete set of orthogonal eigenstates of an observable, used as axes for expanding any state.
K.The complex vector space, with lengths and angles defined, in which the state vectors of a quantum system live.
L.An operator equal to its own conjugate transpose; its eigenvalues are real and its eigenstates form a complete, orthogonal set.

Part B Questions

1.
A qubit is in the state |ψ⟩ = 0.8|0⟩ − 0.6i|1⟩. Which expression is its bra ⟨ψ|?Circle one
1. A0.8⟨0| − 0.6i⟨1|
2. B0.8⟨0| + 0.6i⟨1|
3. C0.8|0⟩ + 0.6i|1⟩
4. D0.6⟨0| − 0.8i⟨1|
2.
Which of these two-by-two matrices is Hermitian?Circle one
1. ARows (1, i) and (i, 1)
2. BRows (0, 1) and (−1, 0)
3. CRows (i, 0) and (0, 1)
4. DRows (2, 1 + i) and (1 − i, −3)
3.
The operator for some observable has eigenvalues −2, 0 and +5. What can a single measurement of that observable return?Circle one
1. AAny value between −2 and +5
2. BOnly the average of the three, which is 1
3. COnly −2, 0 or +5
4. DOnly +5, the largest eigenvalue
4.
Applying the matrix σz to a spin state produces the state the electron is left in after its spin is measured along z.Circle one
TrueFalse
5.
For any two square matrices A and B of the same size, AB = BA.Circle one
TrueFalse
6.
If a system is in an eigenstate of Â with eigenvalue a, then ⟨A⟩ = a and the spread ΔA is zero.Circle one
TrueFalse
7.
An operator that equals its own conjugate transpose, and therefore has only real eigenvalues, is called .Fill in the blank
8.
Two observables whose operators commute, so that they can have definite values at the same time, are called .Fill in the blank
9.
Using commutators, explain why no electron can have a definite spin component along both the x-axis and the z-axis at the same time.Short answer
10.
Describe in words what the momentum operator p̂ = −iħ d/dx does to a wavefunction. Why is the plane wave e^(ikx) a state of definite momentum while the standing wave sin(kx) is not?Short answer
11.
A qubit is prepared in a state proportional to |0⟩ + (1 + i)|1⟩. (a) Normalize it. (b) Find the probabilities of the results 0 and 1. (c) Find ⟨σz⟩. (d) Find ⟨σx⟩ using the sandwich ⟨ψ|σx|ψ⟩.Show your working
Working
Answer:
12.
An electron's spin state gives spin-up along z with probability 0.90. Find (a) ⟨Sz⟩ and (b) the spread ΔSz, both in units of ħ and in J·s.Show your working
Working
Answer:
13.
Multiply out the Pauli matrices to find σxσy and σyσx, and show that [σx, σy] = 2iσz.Show your working
Working
Answer:
14.
An electron is described by a plane wave of wavelength 0.50 nm. Find (a) its wavenumber k, (b) its momentum eigenvalue ħk, and (c) its kinetic energy in eV.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The Hadamard workshop. The Hadamard matrix H is (1/√2) times the matrix with rows (1, 1) and (1, −1). (a) Show that H is Hermitian and that H² is the identity. (b) Use H² = 1 to prove that its eigenvalues can only be +1 or −1, and find the normalized eigenvector for +1. (c) A qubit is prepared in |0⟩ and the observable H is measured. Find the probabilities of +1 and −1 and the expectation value ⟨H⟩, and check ⟨H⟩ with the sandwich ⟨0|H|0⟩. (d) Compute the commutator [H, σz]. Can H and σz have definite values at the same time?

Working

Memory aid

Mnemonic Spell BRA-KET, the notation itself, to recall the whole chapter. B: Bra, the ket laid on its side with every amplitude complex-conjugated. R: Real results only, because observables are Hermitian and Hermitian eigenvalues are real. A: Average, the expectation value ⟨ψ|Â|ψ⟩, a long-run mean rather than a promised result. K: Ket, the state itself, a column of complex amplitudes of total length one. E: Eigenstates, the definite-value states; expand in them, take overlaps, square for probabilities. T: Turn order matters, ÂB̂ − B̂Â, with x̂p̂ − p̂x̂ = iħ behind the uncertainty principle. To use it, walk the letters whenever you meet a measurement problem: B and K, write down the state and its bra; R, check that the operator is Hermitian and find its real eigenvalues; E, expand in the eigenstates and square the overlaps; A, average the results; T, ask whether a second observable commutes with the first.

In one line States are kets, observables are Hermitian operators with real eigenvalues as results, probabilities are squared overlaps, and order matters: x̂p̂ − p̂x̂ = iħ.