Quantum foundations: specify the state and the measurement
Probabilities depend on the measurement basis, and a list of probabilities in one basis does not identify every property of a state. Work with a small, fully specified model before attaching an interpretation to a measurement story.
Course topic: Quantum Mechanics
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
On this page
Scope and responsibility
A finite-dimensional introduction to normalized states, two spin observables and a specified ideal projective-measurement protocol. Freericks is the open 2026 operator-based first edition, Griffiths–Schroeter the 2018 third edition, Sakurai–Napolitano the 2020 third edition. The examples are invented mathematical models of spin one-half, not reports of apparatus performance or a complete interpretation of quantum measurement.
Choose your starting point
I need an algebra-first free route
Choose Freericks for a self-contained operator route; calculus-free still demands sustained algebra and matrix practice.
I need a conventional first quantum course
Choose Griffiths–Schroeter for wave mechanics, formalism and standard calculus-based preparation.
I already have a full undergraduate course
Choose Sakurai–Napolitano to deepen state-space, operators and mixed-state reasoning at graduate level.
Compare the three books
Quantum Mechanics Done Right: The Shortest Path from Novice to Researcher
James K. Freericks · Edition: 1 · 2026
Free operator-based route
- Choose it when
- High-school algebra and trigonometry are stronger than your calculus background.
- Use another route when
- Your immediate syllabus requires conventional scattering or perturbation theory, which this route omits.
- Selected reading
- Introduction to the Quantum World, Mathematical Interlude, then Spin One-Half and Polarized Light.
- Limits and trade-offs
- The complete book is long and extends well beyond a first course. Free reading has CC BY-NC-ND terms and third-party exceptions.
Try before choosing
Normalize the vector (1,2) and compute its overlap with (1,1)/sqrt(2).
Check your work: Norm is sqrt(5); the overlap is 3/sqrt(10), whose squared magnitude is 9/10.
Introduction to Quantum Mechanics
David J. Griffiths; Darrell F. Schroeter · Edition: 3 · 2018
Conventional undergraduate spine
- Choose it when
- You can use calculus, differential equations and linear algebra for wave functions and observables.
- Use another route when
- You need an algebra-only entry or want to skip normalized wave-function reasoning.
- Selected reading
- The Wave Function, Formalism and the spin/angular-momentum material; use the linear-algebra appendix as needed.
- Limits and trade-offs
- A standard course connects finite states with wave mechanics but needs mathematical preparation beyond this two-state case.
Try before choosing
State why the z probabilities alone do not give the x probabilities, using the worked vector.
Check your work: Add amplitudes through an inner product before squaring; adding z probabilities gives no x distribution.
Modern Quantum Mechanics
J. J. Sakurai; Jim Napolitano · Edition: 3 · 2020
Graduate state-space treatment
- Choose it when
- You have prior quantum mechanics and want a precise operator/ensemble language.
- Use another route when
- This is your first encounter with complex vectors or the Born rule.
- Selected reading
- Fundamental Concepts for states and observables; the density-operator material for mixtures.
- Limits and trade-offs
- Abstract notation is efficient for prepared readers but does not replace the first course the preface assumes.
Try before choosing
For diag(1/5,4/5), compute trace and trace of the square, and compare with the pure-state projector.
Check your work: Trace is 1 and trace(rho²)=17/25, whereas a normalized pure projector has trace(rho²)=1.
Check your preparation
Can I normalize and use an inner product?
The squared norm is the sum of squared magnitudes, not the square of a sum. For (1,2), it is 5.
Are the two measurements made on the same copy?
The main case uses separately prepared identical copies for x and z. A sequence on one copy requires its own state-update protocol.
What does hbar do here?
The amplitudes and probabilities are dimensionless; spin outcomes have units of angular momentum and are ±hbar/2.
A six-step study plan
Steps completed: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
-
Define the state space
Selected reading: Freericks Mathematical Interlude or the chosen formalism unit.
What to do: Write the orthonormal basis, normalized state and both observable matrices.
Check your work: Both matrices are Hermitian with eigenvalues ±hbar/2.
-
Find one measurement distribution
Selected reading: States, observables and the Born rule.
What to do: Compute z probabilities and expectation by two methods.
Check your work: The probabilities sum to one; the matrix expectation agrees with the weighted outcomes.
-
Change the basis
Selected reading: Spin one-half material.
What to do: Find normalized x eigenvectors and compute the two overlaps.
Check your work: Square amplitudes after taking the inner products.
-
Check physical dimensions
Selected reading: The same spin material.
What to do: Report expectations with hbar and explain which copies are measured.
Check your work: No claim is made that both noncommuting observables have predetermined outcomes in one run.
-
Insert a defined measurement
Selected reading: Density operators or the stated ensemble rule.
What to do: Solve the follow-up after the z outcome is discarded, including the two joint probabilities.
Check your work: A mixture is averaged at the probability level, not by adding its state vectors.
-
Diagnose lost information
Selected reading: Your two protocols.
What to do: Compare the coherent state and the diagonal ensemble in both bases.
Check your work: They agree in z but differ in x; one-basis probabilities do not fully identify the state.
Worked exercise
One state, two measurement bases
Let |0>,|1> be orthonormal and |psi>=(|0>+2|1>)/sqrt(5). Define Sz=(hbar/2)diag(1,−1) and Sx=(hbar/2)[[0,1],[1,0]]. On separate sets of identically prepared copies, find each outcome probability and expectation.
Assumptions
Ideal normalized two-dimensional spin model, Born probabilities, no time evolution or additional interactions. Each measurement in the main case starts from a fresh copy of psi; it is not a sequential x–z experiment.
- The norm is (1+4)/5=1. Measuring Sz gives +hbar/2 with probability 1/5 and −hbar/2 with probability 4/5; expectation is (hbar/2)(1/5−4/5)=−3hbar/10.
- Normalized Sx eigenvectors are |+x>=(|0>+|1>)/sqrt(2) and |−x>=(|0>−|1>)/sqrt(2). Their overlaps with psi are 3/sqrt(10) and −1/sqrt(10).
- Squared magnitudes give probabilities 9/10 and 1/10. Thus <Sx>=(hbar/2)(9/10−1/10)=2hbar/5. Direct matrix multiplication gives the same value.
Answer and interpretation
Sz: probabilities (1/5,4/5), expectation −3hbar/10. Sx: probabilities (9/10,1/10), expectation 2hbar/5, in positive/negative outcome order.
Test the result
Both distributions sum to one and both expectations lie within [−hbar/2,hbar/2]. Normalizing by 3 instead of sqrt(5), or squaring coefficients before changing basis, fails these calculations.
Change an assumption
A nonselective measurement produces a different ensemble
Keep the original preparation, but before measuring Sx perform an ideal projective Sz measurement on every copy. Retain all copies and discard the outcome record. Use the specified update: outcome + prepares |0>, outcome − prepares |1>, with Born weights. Find the Sx distribution and joint probabilities P(z+,x+), P(z−,x+).
- The first measurement makes a classical ensemble of |0> with weight 1/5 and |1> with weight 4/5. Its density matrix is rho=diag(1/5,4/5); no coherent cross term is retained under the stated update.
- For either |0> or |1>, an x measurement gives + with probability 1/2 and − with probability 1/2. Weighted averaging therefore gives P(x+)=P(x−)=1/2 and <Sx>=0.
- The joint probabilities are P(z+,x+)=(1/5)(1/2)=1/10 and P(z−,x+)=(4/5)(1/2)=2/5. They add to 1/2.
Answer and interpretation
After the inserted measurement, the x outcomes are equally likely. The two requested joint probabilities are 1/10 and 2/5.
Test the result
The original coherent preparation gave P(x+)=9/10. Both preparations have the same z probabilities, but different off-diagonal coherence. Do not reconstruct a pure superposition by adding sqrt-weighted vectors from the mixed ensemble; postselecting one z outcome would also define a different protocol.
Common mistakes
Squaring too early
Changing basis combines amplitudes before the Born square; probabilities cannot be substituted for amplitudes.
Leaving the protocol unstated
Independent-copy comparisons and sequential measurements are different experiments.
Treating a mixture as a state vector
An ensemble combines probabilities or density operators; it is not a coherent sum merely because its weights are known.
Before moving on
- Normalize a vector and compute a distribution in two bases.
- State exactly how an inserted measurement updates the ensemble.
- Explain why equal probabilities in one basis do not imply identical states.
Sources and edition checks
Sources checked:
- Springer: operator-based open first edition
Springer identifies the 2026 ebook, operator-first scope and limited reuse terms; the selected early units develop the needed algebra.
- Cambridge: third-edition quantum front matter
Cambridge’s official front matter verifies the 2018 third edition and wave-function/formalism progression.
- Cambridge: Modern Quantum Mechanics third edition
Cambridge describes the 2020 third edition as senior/first-year graduate, so it is a continuation rather than a beginner substitute.
These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.