Bases and Representations

A basis supplies coordinates for abstract quantum objects. Once a basis is chosen,

• a ket becomes a column of complex components;
• a bra becomes the corresponding conjugate-transpose row;
• an operator becomes a matrix, differential expression, or integral kernel;
• a density operator becomes a matrix or kernel;
• selected components become measurement amplitudes when the basis matches a projective measurement.

The central rule is:

The state is basis independent; its coordinates are basis dependent.

The same rule applies to operators and density operators. A matrix is not an operator without a declared basis, just as a column is not an abstract state without a coordinate convention. Physical predictions remain invariant when every represented object is transformed consistently.

Abstract object	Representation in an orthonormal basis
state vector $\lvert\psi\rangle$	components $c_n=\langle e_n\rvert\psi\rangle$
bra $\langle\psi\rvert$	row $c^\dagger$
operator $A$	matrix $(A_E)_{mn}=\langle e_m\rvert A\lvert e_n\rangle$
density operator $\rho$	matrix $(\rho_E)_{mn}=\langle e_m\rvert\rho\lvert e_n\rangle$
inner product	$d^\dagger c$
expectation value	$c^\dagger A_Ec$

Here $E$ denotes the chosen ordered orthonormal basis. Changing $E$ changes the entries, not the abstract objects or correctly computed predictions.

What a Basis Is

In a $d$-dimensional complex vector space, an ordered set

$$E=(|e_1\rangle,\ldots,|e_d\rangle)$$

is a basis if its vectors are linearly independent and span the space. Every vector then has one and only one expansion

$$|\psi\rangle = \sum_{n=1}^{d}c_n|e_n\rangle.$$

The order matters because it fixes which coordinate occupies each row of the column

$$[\psi]_E = \begin{pmatrix} c_1\\ \vdots\\ c_d \end{pmatrix}.$$

Reordering the basis reorders the coordinates. Rephasing one basis vector changes the phase of its coordinate. Neither operation changes the abstract state.

For an infinite-dimensional Hilbert space, a Hilbert basis means a complete orthonormal set. A vector may require a convergent infinite expansion rather than a finite linear combination. In a separable Hilbert space, the basis can be chosen countable:

$$|\psi\rangle = \sum_{n=1}^{\infty}c_n|e_n\rangle,$$

where the partial sums converge in the Hilbert-space norm.

Orthonormality and Completeness

An orthonormal basis satisfies

$$\langle e_m|e_n\rangle = \delta_{mn}.$$

Orthonormality makes coefficient extraction especially simple. Completeness gives the resolution of the identity

$$I = \sum_n|e_n\rangle\langle e_n|,$$

where an infinite sum is understood through norm convergence on vectors. Thus

$$\begin{aligned} |\psi\rangle &= I|\psi\rangle \\ &= \sum_n |e_n\rangle \langle e_n|\psi\rangle. \end{aligned}$$

The expansion coefficients are therefore

$$c_n = \langle e_n|\psi\rangle.$$

For a normalized state, Parseval’s identity gives

$$\sum_n|c_n|^2 = \langle\psi|\psi\rangle = 1.$$

More generally, for any two vectors,

$$\langle\phi|\psi\rangle = \sum_n \langle\phi|e_n\rangle \langle e_n|\psi\rangle.$$

An orthonormal set that is not complete resolves only the projector onto its closed span, not the identity on the whole Hilbert space.

The mathematical notions of total sets, convergence, and completeness are developed in Completeness and Orthonormal Bases.

Abstract State Versus Coordinate Column

Fix an ordered orthonormal basis $E$. The coordinate map is

$$C_E:\mathcal H\longrightarrow\mathbb C^d, \qquad C_E|\psi\rangle=[\psi]_E.$$

In a countably infinite basis, the target is the sequence space $\ell^2$. The map is linear:

$$C_E \left( a|\psi\rangle+b|\phi\rangle \right) = a[\psi]_E+b[\phi]_E.$$

Because the basis is orthonormal, it preserves the inner product:

$$\langle\phi|\psi\rangle = [\phi]_E^\dagger[\psi]_E.$$

Thus $C_E$ is a unitary identification of the abstract Hilbert space with its coordinate model. The identification is useful but not canonical: a different basis gives a different map.

The same column can represent different abstract vectors under different basis declarations. For example,

$$\begin{pmatrix}1\\0\end{pmatrix}$$

represents $|+z\rangle$ in the ordered $z$-basis and $|+x\rangle$ in the ordered $x$-basis. A coordinate column without its basis label is incomplete information.

Bras and Inner Products in Coordinates

If

$$|\psi\rangle = \sum_n c_n|e_n\rangle,$$

then the corresponding bra is

$$\langle\psi| = \sum_n c_n^*\langle e_n|.$$

In coordinates, the ket is $c$ and the bra is $c^\dagger$. For

$$|\phi\rangle = \sum_n d_n|e_n\rangle,$$

the inner product is

$$\langle\phi|\psi\rangle = d^\dagger c = \sum_n d_n^*c_n.$$

This expression uses the site’s convention that the inner product is conjugate-linear in the bra argument and linear in the ket argument. A coordinate calculation that omits the complex conjugation is not an inner product.

Expansion Coefficients as Measurement Amplitudes

Suppose the basis vectors are the normalized eigenstates of a nondegenerate projective measurement. The outcome projectors are

$$P_n = |e_n\rangle\langle e_n|.$$

For

$$|\psi\rangle = \sum_n c_n|e_n\rangle,$$

the component $c_n$ is the outcome amplitude:

$$c_n = \langle e_n|\psi\rangle.$$

The Born probability is

$$p(n) = \langle\psi|P_n|\psi\rangle = |c_n|^2.$$

This interpretation requires more than a coordinate expansion. It requires that the basis correspond to the measurement being performed. The components of $|\psi\rangle$ in the energy basis give energy amplitudes; its components in a spin-$x$ basis give spin-$x$ amplitudes. One component list cannot be squared indiscriminately to answer every measurement question.

For a degenerate outcome $a$, the physical effect is the projector $P_a$ onto the entire eigenspace. Any orthonormal basis $\{|a,r\rangle\}_{r=1}^{g_a}$ within that subspace gives

$$P_a = \sum_{r=1}^{g_a} |a,r\rangle\langle a,r|,$$

and

$$p(a) = \sum_{r=1}^{g_a} |\langle a,r|\psi\rangle|^2.$$

Changing the basis inside the degenerate eigenspace changes the individual components but not their sum or the outcome probability.

Operator Matrices

Let $A$ be a linear operator. In the orthonormal basis $E$, its matrix entries are

$$(A_E)_{mn} = \langle e_m|A|e_n\rangle.$$

The matrix reconstructs the operator through

$$A = \sum_{m,n} |e_m\rangle (A_E)_{mn} \langle e_n|.$$

If $c=[\psi]_E$, then

$$C_E(A|\psi\rangle) = A_Ec.$$

Indeed, the $m$th output component is

$$\begin{aligned} \langle e_m|A|\psi\rangle &= \sum_n \langle e_m|A|e_n\rangle c_n \\ &= \sum_n(A_E)_{mn}c_n. \end{aligned}$$

The expectation value becomes

$$\langle A\rangle_\psi = \langle\psi|A|\psi\rangle = c^\dagger A_Ec.$$

An operator can be diagonal in one basis and dense in another. Diagonality is a representational property tied to an eigenbasis. Eigenvalues and abstract eigenspaces are basis independent.

The concrete matrix, differential, multiplication, and kernel forms are developed in Operator Representations.

Density-Operator Matrices

A density operator $\rho$ has matrix entries

$$(\rho_E)_{mn} = \langle e_m|\rho|e_n\rangle.$$

In this basis, the diagonal entries are the probabilities for the projective measurement defined by $E$:

$$p(n) = (\rho_E)_{nn}.$$

The off-diagonal entries encode phase coherence relative to the chosen basis. Their numerical values are basis dependent. A density operator that is diagonal in one basis need not be diagonal in another.

For an effect $F$, the probability rule is represented as

$$p(F) = \operatorname{Tr}(\rho_EF_E).$$

The trace is unchanged by a consistent basis transformation. Positivity, rank, spectrum, purity, and entropy are also invariant, even though individual matrix entries change.

For a pure state with component column $c$,

$$\rho_E = cc^\dagger.$$

Its entries are

$$(\rho_E)_{mn} = c_mc_n^*.$$

This is the coordinate version of $\rho=|\psi\rangle\langle\psi|$.

Passive Change of Orthonormal Basis

Let the old ordered orthonormal basis be $E=(|e_n\rangle)$ and the new one be $F=(|f_a\rangle)$. This page follows the repository convention

$$S_{an} = \langle f_a|e_n\rangle.$$

If

$$c_n=\langle e_n|\psi\rangle, \qquad d_a=\langle f_a|\psi\rangle,$$

then

$$d_a = \sum_nS_{an}c_n,$$

or

$$d=Sc.$$

Completeness and orthonormality imply that $S$ is unitary:

$$SS^\dagger = S^\dagger S = I.$$

Operator and density matrices transform with the matching rule

$$A_F = SA_ES^\dagger, \qquad \rho_F = S\rho_ES^\dagger.$$

Consequently,

$$d^\dagger A_Fd = c^\dagger A_Ec,$$

and

$$\operatorname{Tr}(\rho_FF_F) = \operatorname{Tr}(\rho_EF_E).$$

Some texts define the basis-change matrix in the inverse direction, placing the daggers differently. Either convention is valid. Mixing conventions inside one calculation is not.

The full derivation, invariant checks, and active-versus-passive bookkeeping belong to Change of Basis.

Coordinate Change Versus Physical Transformation

A passive basis change leaves the abstract state fixed:

$$|\psi\rangle \text{ fixed}, \qquad c\longmapsto d=Sc.$$

An active unitary transformation changes the state relative to a fixed basis:

$$|\psi\rangle \longmapsto U|\psi\rangle.$$

The same numerical unitary matrix may appear in both calculations, but the physical statements differ. In the passive case, every represented object is rewritten and all predictions are unchanged. In the active case, the state or apparatus is physically transformed, and predictions relative to a fixed measurement can change.

There is a third distinction: changing the measurement basis while keeping the prepared state fixed. That is a change of experimental question, not a mere coordinate rewrite. The new outcome probabilities must be computed from the new projectors. This workflow is developed in Probability in Different Bases.

Worked Spin Example: z and x Bases

For spin-$1/2$, use the ordered bases

$$E_z=(|+z\rangle,|-z\rangle),$$

and

$$E_x=(|+x\rangle,|-x\rangle),$$

where

$$\begin{aligned} |+x\rangle &= \frac{|+z\rangle+|-z\rangle}{\sqrt2}, \\ |-x\rangle &= \frac{|+z\rangle-|-z\rangle}{\sqrt2}. \end{aligned}$$

With rows labeled by $+x,-x$ and columns by $+z,-z$, the overlap matrix is

$$S_{xz} = \frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}.$$

For the abstract state

$$|\psi\rangle = \alpha|+z\rangle + \beta|-z\rangle,$$

the $z$-basis coordinate column is

$$c_z = \begin{pmatrix} \alpha\\ \beta \end{pmatrix},$$

while the $x$-basis column is

$$\begin{aligned} c_x &= S_{xz}c_z \\ &= \frac1{\sqrt2} \begin{pmatrix} \alpha+\beta\\ \alpha-\beta \end{pmatrix}. \end{aligned}$$

For $|\psi\rangle=|+z\rangle$,

$$c_z = \begin{pmatrix}1\\0\end{pmatrix}, \qquad c_x = \frac1{\sqrt2} \begin{pmatrix}1\\1\end{pmatrix}.$$

The state has not changed. The second column shows that an $S_x$ measurement has equal outcome probabilities.

For $|\psi\rangle=|+x\rangle$,

$$c_z = \frac1{\sqrt2} \begin{pmatrix}1\\1\end{pmatrix}, \qquad c_x = \begin{pmatrix}1\\0\end{pmatrix}.$$

The same state is a two-component superposition in the $z$-basis and one basis vector in the $x$-basis. This is why superposition language must name a basis or decomposition.

Worked Spin Example: Operator Matrices

In the $z$-basis,

$$(\sigma_z)_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, \qquad (\sigma_x)_z = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}.$$

In the $x$-basis,

$$\begin{aligned} (\sigma_z)_x &= S_{xz}(\sigma_z)_zS_{xz}^\dagger \\ &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \end{aligned}$$

whereas

$$(\sigma_x)_x = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

The abstract operators have not exchanged identities. Rather, the matrix that represents $\sigma_z$ in the $x$-basis happens to equal the matrix that represents $\sigma_x$ in the $z$-basis. A bare matrix cannot say which operator it represents until the basis is declared.

For $|+z\rangle$, the expectation value of $\sigma_z$ is one in either coordinate system:

$$\begin{aligned} c_z^\dagger(\sigma_z)_zc_z &=1, \\ c_x^\dagger(\sigma_z)_xc_x &=1. \end{aligned}$$

The spin operators and their algebra are developed in Pauli Matrices.

Basis Ordering and Phase Conventions

An ordered basis includes choices that are physically conventional. If

$$|e_n'\rangle = e^{i\beta_n}|e_n\rangle,$$

then the same state has components

$$c_n' = \langle e_n'|\psi\rangle = e^{-i\beta_n}c_n.$$

The operator entries transform as

$$A_{mn}' = e^{-i\beta_m} e^{i\beta_n} A_{mn}.$$

Coefficient phases therefore cannot be interpreted without the corresponding basis-phase convention. Probabilities and expectation values remain unchanged when all entries are transformed consistently.

Permuting the basis acts similarly with a permutation matrix. A statement such as “the state is the first standard column” depends on both basis vectors and ordering.

Nonorthogonal Bases and Dual Bases

Not every useful basis is orthonormal. Let

$$B=(|b_1\rangle,\ldots,|b_d\rangle)$$

be a linearly independent basis. Its Gram matrix is

$$G_{jk} = \langle b_j|b_k\rangle.$$

$G$ is Hermitian and positive definite. Define the dual bras by

$$\langle b^j| = \sum_k(G^{-1})_{jk}\langle b_k|.$$

They satisfy

$$\langle b^j|b_k\rangle = \delta^j_{\ k}.$$

The resolution of the identity is now

$$I = \sum_j|b_j\rangle\langle b^j|.$$

Therefore

$$|\psi\rangle = \sum_j c^j|b_j\rangle, \qquad c^j = \langle b^j|\psi\rangle.$$

The direct overlaps

$$a_j = \langle b_j|\psi\rangle$$

are not the expansion coefficients. They obey

$$a_j = \sum_kG_{jk}c^k.$$

For coefficient columns $c$ and $d$, the inner product becomes

$$\langle\phi|\psi\rangle = d^\dagger Gc.$$

This reduces to $d^\dagger c$ only when $G=I$.

Nonorthogonal basis vectors also cannot serve as distinct outcomes of one projective measurement. Their expansion coefficients are not automatically exclusive outcome amplitudes. Nonorthogonal state discrimination requires the broader POVM language.

Incomplete Sets and Subspace Projections

Suppose $\{|e_n\rangle\}_{n=1}^{N}$ is an orthonormal set but not a complete basis for $\mathcal H$. Then

$$P_N = \sum_{n=1}^{N}|e_n\rangle\langle e_n|$$

is the orthogonal projector onto its span. The truncated expansion is

$$|\psi_N\rangle = P_N|\psi\rangle = \sum_{n=1}^{N}c_n|e_n\rangle.$$

For a normalized state, the discarded norm is

$$\epsilon_N = 1-\langle\psi_N|\psi_N\rangle = \sum_{n>N}|c_n|^2.$$

If $\epsilon_N<1$, the normalized approximation is

$$|\widetilde\psi_N\rangle = \frac{|\psi_N\rangle} {\sqrt{1-\epsilon_N}}.$$

Its fidelity with the exact state is

$$|\langle\psi|\widetilde\psi_N\rangle|^2 = 1-\epsilon_N.$$

This gives a useful state-norm diagnostic. It does not by itself control the expectation of every unbounded operator; high-energy tails can matter strongly for derivatives, kinetic energy, or singular observables.

Continuous Representations

Position and momentum notation resembles a basis expansion but uses generalized eigenvectors. Formally,

$$I = \int_{-\infty}^{\infty} |x\rangle\langle x|\,dx,$$

and

$$\langle x|x'\rangle = \delta(x-x').$$

The position-space wavefunction is the continuous component function

$$\psi(x) = \langle x|\psi\rangle.$$

The abstract state is represented as

$$|\psi\rangle = \int_{-\infty}^{\infty} \psi(x)|x\rangle\,dx,$$

with the integral interpreted through spectral or distributional methods. Inner products become

$$\langle\phi|\psi\rangle = \int_{-\infty}^{\infty} \phi^*(x)\psi(x)\,dx.$$

The momentum representation is

$$\widetilde\psi(p) = \langle p|\psi\rangle.$$

With the site’s Fourier convention,

$$\widetilde\psi(p) = \frac1{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar}\psi(x)\,dx.$$

Position and momentum wavefunctions are two representations of one state, not two states. Their norms agree because the Fourier transform is unitary.

The symbols $|x\rangle$ and $|p\rangle$ are not normalizable Hilbert-space basis vectors. They are generalized eigenkets associated with a continuous spectral representation. The disciplined mathematical setting is introduced in Rigged Hilbert Spaces: First Look.

Discrete, Continuous, and Mixed Spectra

Not every representation is purely a sum or purely an integral. A self-adjoint operator can have discrete and continuous spectral parts. Formal expansions may therefore combine both:

$$|\psi\rangle = \sum_n c_n|n\rangle + \int d\lambda\, c(\lambda)|\lambda\rangle.$$

The rigorous basis-independent object is the spectral measure of the operator. Generalized eigenvectors are useful coordinate notation when a suitable spectral representation exists. They do not replace questions of self-adjointness, domains, or measure normalization.

Choosing a Useful Representation

All complete orthonormal bases are equivalent as coordinate systems, but they are not equally convenient for a calculation.

• An eigenbasis of the measured observable makes outcome amplitudes explicit.
• An energy basis diagonalizes a time-independent Hamiltonian.
• Position representation makes local potentials multiplicative.
• Momentum representation diagonalizes free-particle motion.
• Symmetry-adapted bases organize irreducible sectors and selection rules.
• Product bases expose subsystem labels; Schmidt bases expose bipartite pure entanglement.
• Localized bases can make sparse couplings or boundary conditions transparent.

Convenience does not create physical privilege. A basis becomes physically distinguished only through additional structure such as a Hamiltonian, measurement interaction, symmetry, boundary condition, or decohering environment.

Numerical Representations and Truncation

Computations replace an infinite Hilbert space by a finite representation. That choice is part of the model and must be tested.

For a truncated basis, report at least:

• the basis functions and their ordering;
• the cutoff rule and retained dimension;
• units and normalization conventions;
• boundary conditions or box size, when present;
• convergence of target observables as the cutoff changes;
• the discarded norm or another controlled error diagnostic;
• whether the represented operators preserve the truncated subspace.

An ill-conditioned nonorthogonal basis can amplify roundoff error because its Gram matrix has eigenvalues on very different scales. Near-linear dependence should be diagnosed through singular values or Gram-matrix conditioning, not hidden by unstable inversion.

Sparse and dense matrices are computational storage choices, not different operators. Likewise, a grid wavefunction, spectral coefficient vector, and tensor-network parameterization can represent the same state approximately with different error structures.

What Is Basis Invariant

Under a passive unitary basis change, the following remain unchanged:

• vector norms and inner products;
• transition probabilities;
• expectation values and variances;
• operator eigenvalues and multiplicities;
• traces and determinants in finite dimension;
• density-operator positivity, rank, spectrum, purity, and entropy;
• commutation relations when every operator is transformed consistently;
• physical projectors onto degenerate eigenspaces;
• whether a state is pure or mixed.

Individual components, matrix entries, diagonality, sparsity, and the number of nonzero displayed coefficients are generally basis dependent.

Entanglement has an additional qualification. It is invariant under local basis changes once a tensor-product decomposition is fixed, but it can depend on which subsystem decomposition is regarded as physical.

A Representation Audit

Before trusting a coordinate calculation, ask:

1. What is the abstract Hilbert space? State dimensions, boundary conditions, and subsystem structure.
2. What is the ordered basis or representation? Include phase, ordering, units, and normalization conventions.
3. Is the set complete and orthonormal? If not, identify the projector or Gram matrix and dual basis.
4. Which object is being represented? Distinguish ket, operator, density operator, effect, and coordinate map.
5. Does the basis match the measurement? Components are outcome amplitudes only in the relevant orthonormal measurement basis.
6. Were all objects transformed consistently? State columns and operator matrices must use the same basis.
7. Is the change passive or active? A coordinate rewrite and a physical unitary operation answer different questions.
8. Are generalized kets being used? Replace naive Hilbert-space sums with the appropriate spectral or distributional interpretation.
9. Is the representation truncated? Test convergence and sensitivity to the cutoff.

Common Mistakes

Writing a column without naming the basis

The entries alone do not identify an abstract vector. Attach a basis label or declare the convention once and maintain it.

Calling superposition an absolute property

A state that has many components in one basis can be one basis vector in another. Name the observable, modes, paths, or decomposition.

Transforming only the state

A passive change of coordinates also transforms operator and density matrices. Mixing representations generally changes numerical predictions incorrectly.

Confusing coordinate change with measurement change

Rewriting every object leaves probabilities invariant. Rotating the apparatus changes the projectors and can change the probability distribution.

Using orthonormal formulas in a nonorthogonal basis

For a nonorthogonal basis, direct overlaps are not expansion coefficients and inner products contain the Gram matrix.

Treating continuous eigenkets as normalized vectors

Delta normalization is distributional. Exact position and momentum kets do not belong to the ordinary Hilbert space of normalizable states.

Assuming a converged state norm controls every observable

Unbounded operators can remain sensitive to small high-energy components. Check convergence of the actual quantities of interest.

Connections

• State Vectors develops abstract kets before coordinates are chosen.
• Superposition and Relative Phase explains why component structure depends on the basis while interference is physically testable.
• Change of Basis gives the complete passive transformation derivation.
• Wavefunctions as Representations develops position-space coordinates and their probability meaning.
• Operator Representations compares matrices, differential operators, multiplication operators, and kernels.
• Representation Translation Table provides a compact notation crosswalk.

References

1. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958, Chapters I and II.
2. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955, Chapter II.
3. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. I, Wiley, 1977, Chapters II and III.
4. R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, Chapters 1, 4, and 5.
5. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, Chapters 1 and 3.
6. B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013, Chapters 3 and 6.
7. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1980, Chapters II and VII.
8. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, Sections 2.1 and 2.2.
9. G. B. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians, American Mathematical Society, 2008, Appendix A.

Exercises

1. One state in two spin bases. Let

   $$|\psi\rangle = \frac{|+z\rangle+i|-z\rangle}{\sqrt2}.$$

   Find its coordinate columns in the ordered $z$- and $x$-bases. Compute the probabilities for $S_z$ and $S_x$ measurements.

Solution

In the $z$-basis,

$$c_z = \frac1{\sqrt2} \begin{pmatrix} 1\\ i \end{pmatrix}.$$

Using the overlap matrix,

$$\begin{aligned} c_x &= S_{xz}c_z \\ &= \frac12 \begin{pmatrix} 1+i\\ 1-i \end{pmatrix}. \end{aligned}$$

Therefore

$$P(+z)=P(-z)=\frac12,$$

and

$$P(+x) = \frac{|1+i|^2}{4} = \frac12, \qquad P(-x)=\frac12.$$

The phases of the $x$-basis amplitudes differ even though their magnitudes are equal.

2. Consistent operator transformation. In the $z$-basis, let

   $$c_z = \begin{pmatrix} \cos(\theta/2)\\ e^{i\phi}\sin(\theta/2) \end{pmatrix}.$$

   Transform both $c_z$ and $(\sigma_z)_z$ to the $x$-basis and verify that the expectation value remains $\cos\theta$.

Solution

Write $S=S_{xz}$. The transformed objects are

$$c_x=Sc_z, \qquad (\sigma_z)_x = S(\sigma_z)_zS^\dagger.$$

Then

$$\begin{aligned} c_x^\dagger(\sigma_z)_xc_x &= c_z^\dagger S^\dagger S(\sigma_z)_zS^\dagger Sc_z \\ &= c_z^\dagger(\sigma_z)_zc_z \\ &= \cos^2\frac\theta2 - \sin^2\frac\theta2 \\ &= \cos\theta. \end{aligned}$$

The cancellation uses $S^\dagger S=I$. Transforming only the state column would not represent the same expectation-value calculation.

3. Rephasing a basis. Let

   $$|e_1'\rangle=|e_1\rangle, \qquad |e_2'\rangle=e^{i\beta}|e_2\rangle.$$

   If $|\psi\rangle=c_1|e_1\rangle+c_2|e_2\rangle$, find its new components. Derive the new off-diagonal matrix entry $A_{12}'$ and verify that $c^\dagger Ac$ is unchanged.

Solution

The new components are

$$c_1'=c_1, \qquad c_2'=e^{-i\beta}c_2.$$

The off-diagonal operator entry is

$$A_{12}' = \langle e_1'|A|e_2'\rangle = e^{i\beta}A_{12}.$$

Similarly,

$$A_{21}'=e^{-i\beta}A_{21},$$

while the diagonal entries are unchanged. The cross contribution transforms as

$$(c_1')^*A_{12}'c_2' = c_1^* e^{i\beta}A_{12} e^{-i\beta}c_2 = c_1^*A_{12}c_2.$$

The other terms behave similarly, so the expectation value is invariant.

4. A nonorthogonal basis. In $\mathbb C^2$, take

   $$|b_1\rangle=|0\rangle, \qquad |b_2\rangle = \frac{|0\rangle+|1\rangle}{\sqrt2}.$$

   Find the Gram matrix, dual kets, and expansion coefficients of $|1\rangle$ in this basis.

Solution

The Gram matrix and its inverse are

$$G = \begin{pmatrix} 1&1/\sqrt2\\ 1/\sqrt2&1 \end{pmatrix}, \qquad G^{-1} = \begin{pmatrix} 2&-\sqrt2\\ -\sqrt2&2 \end{pmatrix}.$$

The dual kets are

$$|b^1\rangle = |0\rangle-|1\rangle, \qquad |b^2\rangle = \sqrt2|1\rangle.$$

They satisfy $\langle b^j|b_k\rangle=\delta^j_{\ k}$. Hence the expansion coefficients of $|1\rangle$ are

$$c^1 = \langle b^1|1\rangle = -1, \qquad c^2 = \langle b^2|1\rangle = \sqrt2.$$

Indeed,

$$-|b_1\rangle + \sqrt2|b_2\rangle = |1\rangle.$$

The direct overlaps $\langle b_j|1\rangle$ are $0$ and $1/\sqrt2$, so they are not the expansion coefficients.

5. Truncation error. Let

   $$|\psi\rangle = \sum_{n=1}^{\infty}c_n|e_n\rangle$$

   be normalized and let $P_N$ retain the first $N$ basis states. Prove that the fidelity between $|\psi\rangle$ and the normalized truncated state is $1-\epsilon_N$, where

   $$\epsilon_N=\sum_{n>N}|c_n|^2.$$

Solution

The truncated vector is

$$|\psi_N\rangle=P_N|\psi\rangle,$$

with norm

$$\langle\psi_N|\psi_N\rangle = \sum_{n=1}^{N}|c_n|^2 = 1-\epsilon_N.$$

Its normalized form is

$$|\widetilde\psi_N\rangle = \frac{|\psi_N\rangle}{\sqrt{1-\epsilon_N}}.$$

Because $P_N$ is an orthogonal projector,

$$\langle\psi|\psi_N\rangle = \langle\psi|P_N|\psi\rangle = 1-\epsilon_N.$$

Therefore

$$|\langle\psi|\widetilde\psi_N\rangle|^2 = \frac{(1-\epsilon_N)^2}{1-\epsilon_N} = 1-\epsilon_N.$$

6. Degenerate eigenspace. Let an observable have a two-dimensional eigenspace with orthonormal bases $\{|u_1\rangle,|u_2\rangle\}$ and $\{|v_1\rangle,|v_2\rangle\}$. Prove that

   $$\sum_{r=1}^{2}|u_r\rangle\langle u_r| = \sum_{r=1}^{2}|v_r\rangle\langle v_r|.$$

   Explain which probability is basis independent.

Solution

Both sums are the orthogonal projector $P$ onto the same eigenspace. To verify this directly, let

$$|v_r\rangle = \sum_sU_{rs}|u_s\rangle,$$

where $U$ is unitary. Then

$$\begin{aligned} \sum_r|v_r\rangle\langle v_r| &= \sum_{r,s,t} U_{rs}U_{rt}^* |u_s\rangle\langle u_t| \\ &= \sum_{s,t}\delta_{st} |u_s\rangle\langle u_t| \\ &= \sum_s|u_s\rangle\langle u_s|. \end{aligned}$$

For state $|\psi\rangle$, the degenerate outcome probability

$$p=\langle\psi|P|\psi\rangle$$

is basis independent. The individual squared components inside the subspace depend on which internal basis is chosen.

7. Position and momentum representations. Using

   $$\langle x|p\rangle = \frac{e^{ipx/\hbar}}{\sqrt{2\pi\hbar}},$$

   derive the Fourier-transform expression for $\widetilde\psi(p)=\langle p|\psi\rangle$. State the corresponding inverse transform and norm identity.

Solution

Insert the position completeness relation:

$$\begin{aligned} \widetilde\psi(p) &= \langle p|\psi\rangle \\ &= \int_{-\infty}^{\infty} \langle p|x\rangle \langle x|\psi\rangle\,dx \\ &= \frac1{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar}\psi(x)\,dx. \end{aligned}$$

The inverse transform is

$$\psi(x) = \frac1{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{ipx/\hbar}\widetilde\psi(p)\,dp.$$

Unitarity gives Parseval’s identity

$$\int_{-\infty}^{\infty}|\psi(x)|^2\,dx = \int_{-\infty}^{\infty}|\widetilde\psi(p)|^2\,dp.$$

The generalized kets and integrals are understood spectrally or distributionally, not as ordinary uncountable Hilbert-space sums.

8. Same matrix, different operator declaration. In the $z$-basis, $\sigma_x$ has the matrix

   $$M = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}.$$

   In the $x$-basis, the same matrix represents $\sigma_z$. Verify both statements and explain why the matrix alone does not define an abstract operator.

Solution

By definition in the $z$-basis,

$$(\sigma_x)_z=M.$$

For $S=S_{xz}$,

$$\begin{aligned} (\sigma_z)_x &= S(\sigma_z)_zS^\dagger \\ &= \frac12 \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} \\ &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} =M. \end{aligned}$$

The entry array is identical, but the coordinate maps are different. One matrix acts on $z$-basis columns and represents $\sigma_x$; the other acts on $x$-basis columns and represents $\sigma_z$. An abstract operator is defined only after the matrix is paired with its domain, codomain, and basis identifications.