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Coordinates, Completeness, and Changing Bases (a.k.a. “Same vector, different outfit”)

Vectors (and quantum states) are like songs: the song is the same, but you can write it in different musical notation. A basis is your notation system.

In this mini-lesson we’ll connect four ideas that fit together perfectly:

  1. Coordinates in an orthonormal basis
  2. Completeness (resolution of identity)
  3. Change-of-basis matrices (and why they’re unitary)
  4. The quantum-mechanics meaning: same state, different coordinate lists; probabilities depend on measurement basis

1) Coordinates in an orthonormal basis: “How much of each basis vector?”

Suppose you have an orthonormal basis {∣ek⟩}\{|e_k\rangle\}.

  • Orthonormal means:
    ⟨ej∣ek⟩=δjk\langle e_j|e_k\rangle = \delta_{jk}

Any vector (or state) ∣v⟩|v\rangle can be written as a basis expansion:

∣v⟩=∑kck ∣ek⟩|v\rangle = \sum_k c_k\,|e_k\rangle

The magic is: in an orthonormal basis, the coefficient is just a dot product / inner product:

ck=⟨ek∣v⟩c_k = \langle e_k|v\rangle

So ⟨ek∣v⟩\langle e_k|v\rangle literally answers: “what is the component of ∣v⟩|v\rangle along ∣ek⟩|e_k\rangle?”

2) Completeness: the resolution of identity is a “rebuild button”

Here’s a famous-looking formula that’s actually super friendly:

I=∑k∣ek⟩⟨ek∣I = \sum_k |e_k\rangle\langle e_k|

This is called completeness or the resolution of the identity.

Why is it so useful? Because if you apply it to a vector, it rebuilds the vector from its coordinates:

∣v⟩=I∣v⟩=(∑k∣ek⟩⟨ek∣)∣v⟩=∑k∣ek⟩⟨ek∣v⟩⏟ck|v\rangle = I|v\rangle = \left(\sum_k |e_k\rangle\langle e_k|\right)|v\rangle = \sum_k |e_k\rangle\underbrace{\langle e_k|v\rangle}_{c_k}

So that one line contains the whole story:

  • ⟨ek∣v⟩\langle e_k|v\rangle produces the coordinate
  • ∣ek⟩|e_k\rangle puts the coordinate back in the right direction
  • summing over kk reconstructs ∣v⟩|v\rangle

3) Changing bases: same vector, new coordinate list

Now suppose you have two orthonormal bases:

  • Old basis: {∣ek⟩}\{|e_k\rangle\}
  • New basis: {∣fj⟩}\{|f_j\rangle\}

The vector ∣v⟩|v\rangle doesn’t change. But its coordinate list does.

Let

  • ck=⟨ek∣v⟩c_k = \langle e_k|v\rangle be coordinates in the ee-basis
  • dj=⟨fj∣v⟩d_j = \langle f_j|v\rangle be coordinates in the ff-basis

The change-of-basis matrix

Define a matrix UU with entries

Ujk=⟨fj∣ek⟩U_{jk} = \langle f_j|e_k\rangle

Then the coordinates transform as:

d=Ucd = Uc

(Here cc and dd are column vectors of coefficients.)

Why is UU unitary?

Because both bases are orthonormal. That forces

U†U=IandUU†=IU^\dagger U = I \quad \text{and} \quad UU^\dagger = I

That’s the definition of a unitary matrix. Intuitively:

  • unitary changes of basis preserve lengths and angles
  • in quantum mechanics: unitary changes preserve total probability

Worked example 1: Coordinate transform between two orthonormal bases (2D)

Let the old basis be the standard one:

∣e1⟩=(10),∣e2⟩=(01)|e_1\rangle = \begin{pmatrix}1\\0\end{pmatrix},\quad |e_2\rangle = \begin{pmatrix}0\\1\end{pmatrix}

Make a new orthonormal basis by rotating by angle θ\theta:

∣f1⟩=(cos⁡θsin⁡θ),∣f2⟩=(−sin⁡θcos⁡θ)|f_1\rangle = \begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix},\quad |f_2\rangle = \begin{pmatrix}-\sin\theta\\ \cos\theta\end{pmatrix}

Compute the change-of-basis matrix Ujk=⟨fj∣ek⟩U_{jk}=\langle f_j|e_k\rangle. Since these vectors are real, inner products are just dot products:

\langle f_1|e_1\rangle & \langle f_1|e_2\rangle \\ \langle f_2|e_1\rangle & \langle f_2|e_2\rangle \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{pmatrix}$$ Now take a vector with old coordinates $$c = \begin{pmatrix}1\\0\end{pmatrix}$$ That means $$|v\rangle = 1|e_1\rangle + 0|e_2\rangle = |e_1\rangle$$. New coordinates: $$d = Uc = \begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}\cos\theta\\-\sin\theta\end{pmatrix}$$ So in the rotated basis, $$|v\rangle = (\cos\theta)|f_1\rangle + (-\sin\theta)|f_2\rangle$$ Same vector. Different “ingredient list.” --- ### 4) Operators: same operator, different matrix representation Vectors change coordinates as $$d = Uc$$. Operators also “change their clothes.” Let an operator be $$A$$. Its matrix in a basis depends on which basis you use. If $$[A]_e$$ is the matrix in the $$\{|e\rangle\}$$ basis, and $$[A]_f$$ is the matrix in the $$\{|f\rangle\}$$ basis, then $$[A]_f = U\,[A]_e\,U^\dagger$$ Read it as: “change basis on the input side and on the output side.” --- ### Worked example 2: Same operator, new matrix via $$A \mapsto U^\dagger A U$$ Let’s use a classic quantum operator: Pauli-$$Z$$ in the standard basis $$\{|0\rangle,|1\rangle\}$$: $$[Z]_e = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ Now change to the **Hadamard basis**: $$|+\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix},\quad |-\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\end{pmatrix}$$ The unitary that maps old coordinates to new coordinates here is the Hadamard matrix: $$U = H = \frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$$ Since $$H^\dagger = H$$ and $$H^\dagger H = I$$, it’s unitary. Compute the operator matrix in the new basis: $$[Z]_f = U\,[Z]_e\,U^\dagger = H Z H$$ Carrying out the multiplication gives: $$H Z H = \begin{pmatrix}0&1\\1&0\end{pmatrix} = [X]$$ So: - it’s the **same physical operator** (still “measure along Z”) - but in the $$\{|+\rangle,|-\rangle\}$$ coordinate system, its matrix looks like Pauli-$$X$$ That’s not a contradiction—just a reminder that **matrices are representations**, not the operator itself. --- ### 5) Quantum mechanics link: same state, different coordinate vectors (and probabilities depend on basis) In QM, a state $$|\psi\rangle$$ is the real object. Coordinates depend on the chosen basis. If you expand in basis $$\{|e_k\rangle\}$$: $$|\psi\rangle = \sum_k c_k|e_k\rangle, \quad c_k=\langle e_k|\psi\rangle$$ Then the **Born rule** says: $$\Pr(e_k\text{ when measuring in the }e\text{-basis}) = |c_k|^2$$ Switch the measurement basis to $$\{|f_j\rangle\}$$: $$|\psi\rangle = \sum_j d_j|f_j\rangle, \quad d_j=\langle f_j|\psi\rangle$$ Now probabilities are $$\Pr(f_j) = |d_j|^2$$ Same state. Different basis. Different coordinate list. And that basis choice is exactly what a measurement setting is. --- ## Takeaway - Coordinates in an orthonormal basis are $$c_k=\langle e_k|v\rangle$$ and rebuild the vector via $$|v\rangle=\sum_k c_k|e_k\rangle$$. - Completeness $$I=\sum_k |e_k\rangle\langle e_k|$$ is the “rebuild button” that encodes the whole expansion. - Changing between orthonormal bases uses a **unitary** matrix $$U$$, with coordinate change $$d=Uc$$. - Operators keep their identity, but their **matrices** change as $$[A]_f=U[A]_eU^\dagger$$. - In QM, different bases mean different measurement questions—and probabilities come from the coordinates in that chosen basis.
Course
Introductory Non‑Relativistic Quantum Mechanics: Postulates, Ope
12 units57 lessons
Topics
Quantum Physics (Non-relativistic Quantum Mechanics)Mathematical PhysicsLinear AlgebraDifferential Equations / Boundary-Value ProblemsComplex Analysis (foundational tools)
About this course

Develop an intuition-first, problem-solving mastery of non-relativistic quantum mechanics using the postulates and operator formalism. Core topics include states and representations (kets/bras and wavefunctions), normalization and phase, the Born rule and expectation values, measurement as projectors and spectral decomposition, and unitary time evolution via the Schrödinger equation and U(t)=e^{-iHt/ħ}. Apply commutators and uncertainty relations, switch between position and momentum pictures, and solve standard Hamiltonians: 1D wells and barriers (including tunneling and probability current), the harmonic oscillator (ladder operators), angular momentum and spin-1/2, and the hydrogen atom. Gain moderate facility with approximation methods such as time-independent perturbation theory, the variational principle, and optional WKB.