Quantum amplitudes are complex numbers, but time and position are real variables. Most differentiation in the opening quantum mechanics pages is therefore ordinary real-variable calculus with complex values. The other recurring ingredient is the phase \(e^{-itE/\hbar}\). We build both here, then distinguish them from differentiation with respect to a complex variable.
We assume elementary real analysis, the Euclidean metric and topology, and complex arithmetic. The previous chapter supplies finite-dimensional complex inner products, adjoints, and the finite spectral theorem. Our convention is \(\langle x,y\rangle=\sum_j\overline{x_j}y_j\): the first slot is conjugate-linear. Alphabeta’s state geometry definition uses the opposite convention, linear in the first slot. Translate its pairing by \((u,v)_{\mathrm{AB}}=\langle v,u\rangle_{\mathrm{ours}}\). Conjugate symmetry makes this the conjugate of our pairing; norms and squared overlap magnitudes are identical. This chapter supports the framework and quantities page and autonomous dynamics, checked on October 15, 2026.
Six-chapter reading order
3.1. Conjugation, size, and complex-valued calculus
Definition 3.1. The Euclidean meaning of a complex number
Write \(z=x+iy\), with \(x,y\in\mathbb R\) and \(i^2=-1\). Its real part is \(\operatorname{Re}z=x\), its imaginary part is \(\operatorname{Im}z=y\), its conjugate is \(\overline z=x-iy\), and its modulus is \(|z|=\sqrt{x^2+y^2}\). The metric \(d(z,w)=|z-w|\) is the Euclidean metric under \(x+iy\leftrightarrow(x,y)\). Limits and continuity use this metric.
This identification lets us use real analysis without changing its limit definitions. In particular, \(z_n\to z\) exactly when both real components converge: each component difference is bounded by \(|z_n-z|\), and conversely \(|z_n-z|\le|\operatorname{Re}(z_n-z)|+|\operatorname{Im}(z_n-z)|\).
Proposition 3.2. Conjugation and modulus laws
For complex numbers \(z,w\),
\[\overline{z+w}=\overline z+\overline w,\quad \overline{zw}=\overline z\,\overline w,\quad z\overline z=|z|^2,\quad |zw|=|z||w|,\quad |z+w|\le |z|+|w|.\]Also \(|\overline z|=|z|\), and \(z^{-1}=\overline z/|z|^2\) for \(z\ne0\).
Proof.
The conjugation identities follow by expanding real and imaginary parts; multiplying \((x+iy)(x-iy)\) gives \(x^2+y^2\). Thus \(|zw|^2=zw\overline{zw}=|z|^2|w|^2\); take nonnegative square roots. Since \(\operatorname{Re}(z\overline w)\le|z\overline w|=|z||w|\), expansion gives \(|z+w|^2\le(|z|+|w|)^2\), proving the triangle inequality. The final assertions follow from the same formulas.
□Example and counterexample. For \(z=3+4i\), \(\overline z=3-4i\), \(|z|=5\), and \(z^{-1}=(3-4i)/25\). Modulus is not additive: \(|1+(-1)|=0\), whereas \(|1|+|-1|=2\). Conjugation is real-linear but not complex-linear: \(\overline{iz}=-i\overline z\), which differs from \(i\overline z\) when \(z\ne0\).
Definition 3.3. A real parameter and a complex value
For \(f:I\to\mathbb C\), where \(I\) is an open real interval, define
\[f'(t)=\lim_{h\to0,\ h\in\mathbb R}\frac{f(t+h)-f(t)}h.\]If \(f=u+iv\), then \(f'=u'+iv'\) whenever both real derivatives exist; conversely existence of \(f'\) implies their existence. For real coordinates \(x_1,\ldots,x_d\), partial derivatives are defined by varying one real coordinate, and a complex-valued function is \(C^1\) if all first partial derivatives exist and are continuous.
The derivative equivalence follows from componentwise limits. The real sum, product, and quotient rules remain valid with complex values: addition and multiplication of the expansions \(f(t+h)=f(t)+hf'(t)+o(|h|)\) and \(g(t+h)=g(t)+hg'(t)+o(|h|)\) prove the first two, and \(1/g(t+h)-1/g(t)=-(g(t+h)-g(t))/(g(t+h)g(t))\) proves the last when \(g(t)\ne0\). Here \(r(h)=o(|h|)\) means \(|r(h)|/|h|\to0\).
Conjugation commutes with real differentiation because \(h\) is real and conjugation preserves limits. Consequently
\[\frac{d}{dt}|f(t)|^2=\overline{f'(t)}f(t)+\overline{f(t)}f'(t)=2\operatorname{Re}\bigl(\overline{f(t)}f'(t)\bigr).\]Example. For \(f(t)=t+it^2\), \(f'(t)=1+2it\) and \((|f|^2)'=(t^2+t^4)'=2t+4t^3\). The displayed formula gives the same result. The modulus itself need not be differentiable at a zero: for \(f(t)=t\), \(|f(t)|=|t|\) is not differentiable at zero, although \(|f(t)|^2=t^2\) is.
Definition 3.4. Integration along a real interval
For a continuous \(f=u+iv:[a,b]\to\mathbb C\), define its integral by real Riemann integrals:
\[\int_a^b f(t)\,dt=\int_a^b u(t)\,dt+i\int_a^b v(t)\,dt.\]A vector-valued integral in \(\mathbb C^n\) is defined coordinate by coordinate. Reversing the limits reverses the sign.
Complex Riemann sums converge to this value by componentwise convergence. Their triangle inequality gives \(|\int_a^b f|\le\int_a^b|f|\). Taking conjugates of the sums gives \(\overline{\int f}=\int\overline f\), and multiplication by a constant commutes with the integral. The fundamental theorem follows by applying the real theorem to each component: if \(F'=f\) with \(f\) continuous, then \(\int_a^b f=F(b)-F(a)\). Thus integration by parts follows by integrating the product rule.
Example and counterexample. \(\int_0^1(t+it^2)\,dt=1/2+i/3\). Integration is linear over \(\mathbb C\), but modulus does not commute with it: \(\int_{-1}^1t\,dt=0\), whereas \(\int_{-1}^1|t|\,dt=1\). Integrals over all of \(\mathbb R^d\), and integrals of merely measurable functions, need chapter 4.
3.2. Constructing the exponential
Definition 3.5. The complex exponential
A complex series \(\sum_{n=0}^\infty a_n\) is the limit of its partial sums. It is absolutely convergent if \(\sum_n|a_n|\) is finite. Set \(0!=1\) and \(n!=1\cdot2\cdots n\) for positive integers \(n\). Define
\[\exp z=e^z:=\sum_{n=0}^\infty\frac{z^n}{n!}.\]Proposition 3.6. Convergence, addition, and differentiation
The defining series converges absolutely for every \(z\), and uniformly on every disk \(|z|\le R\). Moreover,
\[e^0=1,\quad e^{z+w}=e^ze^w,\quad e^ze^{-z}=1,\quad \overline{e^z}=e^{\overline z},\quad \lim_{h\to0,\ h\in\mathbb C}\frac{e^{z+h}-e^z}{h}=e^z.\]In particular, \(e^z\ne0\), and for a fixed complex \(a\), \(d(e^{at})/dt=ae^{at}\) with real \(t\).
Proof.
Fix \(R\ge0\). For \(n+1\ge2R\), successive majorant terms \(R^n/n!\) have ratio at most \(1/2\). Their tail is bounded by a geometric series, tending to zero independently of \(|z|\le R\). Absolute convergence implies convergence because \(\mathbb C\cong\mathbb R^2\) is complete: tails of partial sums are bounded by tails of the real absolute series.
To justify multiplication of the two series, put \(A_m=|z|^m/m!\), \(B_n=|w|^n/n!\). The absolute sum over pairs outside \(m+n\le N\) is bounded by
\[\left(\sum_{m>N/2}A_m\right)\left(\sum_nB_n\right)+\left(\sum_mA_m\right)\left(\sum_{n>N/2}B_n\right),\]which tends to zero. We may therefore group the product by \(m+n=k\). The binomial formula gives \(\sum_{m=0}^k z^mw^{k-m}/(m!(k-m)!)=(z+w)^k/k!\). This proves the addition law. Set \(w=-z\) for the inverse law. Conjugation passes through partial sums and their limit, proving its identity.
Finally, for \(|h|\le1\),
\[\left|e^h-1-h\right|\le |h|^2\sum_{n=2}^\infty\frac1{n!}.\]Divide by \(|h|\) to obtain \((e^h-1)/h\to1\). The addition law now gives the complex difference quotient at \(z\). Replacing \(h\) by \(ah\) gives the real derivative formula, including \(a=0\). This proof avoids assuming a theorem about differentiation of infinite series.
□Example. \(e^{(2+i)t}\) satisfies \(f'=(2+i)f\) and \(f(0)=1\). It is the unique solution: if \(g'=(2+i)g\), the product rule gives \((e^{-(2+i)t}g(t))'=0\). Its real and imaginary components are constant by the real mean value theorem, so its initial value determines \(g\).
3.3. Phases and Euler’s formula
A unit complex number, or phase factor, is a number of modulus one. Multiplying by one preserves modulus by Proposition 3.2. The unit circle is \(\{x+iy:x^2+y^2=1\}\). We introduce sine and cosine as coordinates of a particular motion on this circle; no trigonometric identities are prerequisites.
Definition 3.7. Sine and cosine
For real \(t\), define \(\cos t=\operatorname{Re}(e^{it})\) and \(\sin t=\operatorname{Im}(e^{it})\). Thus Euler’s formula is
\[e^{it}=\cos t+i\sin t.\]Grouping the absolutely convergent series into even and odd terms gives
\[\cos t=\sum_{n=0}^\infty\frac{(-1)^nt^{2n}}{(2n)!},\qquad \sin t=\sum_{n=0}^\infty\frac{(-1)^nt^{2n+1}}{(2n+1)!}.\]Proposition 3.8. Motion on the unit circle
For real \(s,t\),
\[|e^{it}|=1,\quad \cos^2t+\sin^2t=1,\quad (\cos t)'=-\sin t,\quad (\sin t)'=\cos t,\] \[\cos(s+t)=\cos s\cos t-\sin s\sin t,\quad \sin(s+t)=\sin s\cos t+\cos s\sin t.\]Also \(\cos(-t)=\cos t\), \(\sin(-t)=-\sin t\), and for real \(x,y\), \(|e^{x+iy}|=e^x\).
Proof.
Conjugation and the addition law give \(|e^{it}|^2=e^{it}e^{-it}=1\). Differentiating \(e^{it}\) gives \(ie^{it}=-\sin t+i\cos t\), yielding both real derivative formulas. Multiplying \(e^{is}\) and \(e^{it}\) yields the addition identities; conjugation yields parity. For real \(x\), \(e^x=e^{x/2}e^{x/2}\) is positive because \(e^{x/2}\) is a nonzero real number. Hence \(|e^{x+iy}|=|e^x||e^{iy}|=e^x\). Also \((e^x)'=e^x>0\), so the real mean value theorem makes the real exponential strictly increasing.
□The curve \(t\mapsto(\cos t,\sin t)\) starts at \((1,0)\) with velocity \((0,1)\). Its velocity \((-\sin t,\cos t)\) is perpendicular to its position and has length one. Thus \(t\) measures distance travelled along the unit circle, the usual radian measure of angle. For a continuously differentiable curve, the distance travelled is computed by integrating its speed; here that integral on \([a,b]\) is \(\int_a^b\sqrt{(-\sin t)^2+(\cos t)^2}\,dt=b-a\). Multiplication by \(e^{it}\) acts on \((x,y)\) by the matrix \(\begin{pmatrix}\cos t&-\sin t\\ \sin t&\cos t\end{pmatrix}\); its columns are orthonormal and its determinant is one, so it is an orientation-preserving Euclidean rotation.
Why integrating speed computes arc length
The arc length of a curve \(\gamma:[a,b]\to\mathbb R^2\) is the supremum of the sums of chord lengths \(\sum_j|\gamma(t_j)-\gamma(t_{j-1})|\) over finite partitions. For a continuously differentiable curve the fundamental theorem gives each chord as \(\int_{t_{j-1}}^{t_j}\gamma'(t)\,dt\), so every such sum is at most \(\int_a^b|\gamma'(t)|\,dt\). Conversely, on a partition with small mesh, uniform continuity of \(\gamma'\) makes its difference from its value at any chosen point on each piece at most \(\varepsilon\). The chord differs from that value times the interval length by at most \(\varepsilon\) times the interval length. The chord sum therefore differs from the corresponding Riemann sum for speed by at most \(\varepsilon(b-a)\). Taking meshes tending to zero proves the reverse inequality for the supremum, hence the integral formula.
Why every unit complex number is a phase, and where π enters
Let \(c(t)=\cos t\) and \(s(t)=\sin t\). Since \(c(0)=1\), initially \(c>0\). There must be a positive zero of \(c\): otherwise \(s'=c>0\) for all positive times, so for any small \(a>0\), \(s(t)\ge s(a)>0\) when \(t\ge a\); then \(c'=-s\le-s(a)\) forces \(c\) below zero in finite time. Let \(q\) be the first positive zero. This exists because zeros form a closed set, are absent near zero, and at least one occurs. On \((0,q)\), \(c>0\), \(s>0\), and \(c'<0\). The identity \(c^2+s^2=1\) then gives \((c(q),s(q))=(0,1)\).
The intermediate value theorem and strict decrease make \(c\) cover \([0,1]\) on \([0,q]\). There \(s=\sqrt{1-c^2}\), so the curve covers the first quadrant of the unit circle. The addition identities give \((c(t+q),s(t+q))=(-s(t),c(t))\), covering the other three quadrants in successive intervals. Define \(\pi=2q\). Four such shifts give period \(4q=2\pi\), with \(e^{i\pi}=-1\) and \(e^{i\pi/2}=i\). Every unit complex number is therefore \(e^{it}\). The first four quadrant arcs show that \(e^{it}=1\) on \([0,2\pi)\) only at zero. Reducing any real \(t\) by an integer multiple of \(2\pi\) proves \(e^{it}=1\) exactly when \(t\in2\pi\mathbb Z\). Here \(\mathbb Z\) denotes the integers.
Consequently each nonzero \(z\) has a polar form \(z=re^{i\theta}\), where \(r=|z|>0\); its angle \(\theta\), called an argument, is determined only up to addition of \(2\pi k\), \(k\in\mathbb Z\). This follows by applying the circle parametrization to \(z/|z|\), and the addition law proves the stated ambiguity.
Example and counterexample. \(i=e^{i\pi/2}\), and \(-1=e^{i\pi}\), so opposite phases can cancel: \(1+e^{i\pi}=0\). Purely imaginary exponents give phases, but arbitrary exponents do not: \(|e^{1+i}|=e>1\). The last inequality follows from \(e=\sum_n1/n!>1\).
The quantum framework identifies normalized vectors differing by a common phase as the same pure preparation; see its phase-invariance statement. For vectors, a common phase preserves inner products: \(\langle e^{i\theta}x,e^{i\theta}y\rangle=e^{-i\theta}e^{i\theta}\langle x,y\rangle=\langle x,y\rangle\). Relative phases still matter. In \(\mathbb C^2\), put \(b=(1,1)/\sqrt2\) and \(\psi_\theta=(1,e^{i\theta})/\sqrt2\). Then
\[|\langle b,\psi_\theta\rangle|^2=\left|\frac{1+e^{i\theta}}2\right|^2=\frac{1+\cos\theta}2.\]It equals one at \(\theta=0\) and zero at \(\theta=\pi\). Both vectors have norm one. The arithmetic shows why preserving a vector’s norm does not preserve all overlaps when its components receive different phases.
3.4. Differentiating a complex variable
Definition 3.9. Complex derivative and holomorphic function
For an open \(U\subset\mathbb C\), a function \(f:U\to\mathbb C\) is complex differentiable at \(a\) if
\[f'(a)=\lim_{h\to0,\ h\in\mathbb C\setminus\{0\}}\frac{f(a+h)-f(a)}h\]exists. It is holomorphic on \(U\) if it is complex differentiable at every point of \(U\), and entire if \(U=\mathbb C\). The increment may now approach zero in any direction in the plane.
This is a stronger requirement than differentiating a curve with a real parameter. Complex differentiability says \(f(a+h)=f(a)+ch+o(|h|)\), with the same complex number \(c\) for every direction. It implies continuity by the triangle inequality, and its derivative is unique by uniqueness of a limit. Proposition 3.6 proves that the exponential is entire.
Example and counterexample. \(f(z)=z^2\) is entire since \(((a+h)^2-a^2)/h=2a+h\to2a\). But \(f(z)=\overline z\) has no complex derivative anywhere: its quotient is \(\overline h/h=1\) for real \(h\) and \(-1\) for purely imaginary \(h\). Its restriction to the real line is nevertheless differentiable, with derivative one. A time-dependent amplitude need not be holomorphic in a complex time variable for its real time derivative to exist.
Proposition 3.10. Cauchy–Riemann equations
Write \(f(x+iy)=u(x,y)+iv(x,y)\). Complex differentiability at \(a\) implies the partial derivatives at \(a\) exist and satisfy
\[u_x=v_y,\qquad u_y=-v_x,\qquad f'(a)=u_x+iv_x.\]These are the Cauchy–Riemann equations. Conversely, if \(u,v\) have continuous first partial derivatives near \(a\) and the equations hold at \(a\), then \(f\) is complex differentiable there. If those hypotheses hold throughout an open set, \(f\) is holomorphic there.
Proof.
Approach along real increments \(h\). The limit gives \(f'(a)=u_x+iv_x\). Approach along \(ik\), \(k\in\mathbb R\): the limit gives \(f'(a)=(u_y+iv_y)/i=v_y-iu_y\). Equality of real and imaginary parts yields the equations.
For the converse, continuous partial derivatives give a real linear approximation. To see this directly for \(u\), telescope its increment into a horizontal and a vertical increment. Apply the real mean value theorem to each: the result is \(u_x(a)p+u_y(a)q\), plus an error bounded by \((|p|+|q|)\varepsilon(p,q)\), where continuity of the partials gives \(\varepsilon(p,q)\to0\). Since \(|p|+|q|\le\sqrt2\sqrt{p^2+q^2}\), the error is \(o(\sqrt{p^2+q^2})\). The same holds for \(v\). With \(h=p+iq\), the Cauchy–Riemann equations convert their combined linear part to \((u_x+iv_x)h\). Dividing the remainder by \(h\) proves the complex derivative exists.
□Example. For \(z^2\), \(u=x^2-y^2\) and \(v=2xy\); thus \(u_x=2x=v_y\) and \(u_y=-2y=-v_x\). For \(\overline z\), \(u=x\), \(v=-y\), so \(u_x=1\ne-1=v_y\).
Counterexample to dropping regularity. Define \(f(0)=0\) and \(f(x+iy)=xy/\sqrt{x^2+y^2}\) away from zero, with imaginary part zero. It is continuous at zero because \(|xy|\le(x^2+y^2)/2\). Both partial derivatives there vanish, so the Cauchy–Riemann equations hold there. But along \(h=t+it\), \(t>0\), the quotient is \(1/(\sqrt2(1+i))\), while along the real axis it is zero. Thus partial derivatives and the equations at a point do not alone imply complex differentiability.
The scalar phases and finite-dimensional evolution below require no contour integration or residue theory.
3.5. From scalar phases to matrix evolution
Definition 3.11. Matrix exponential
For an \(n\times n\) complex matrix \(A\), set
\[e^A=\sum_{k=0}^\infty\frac{A^k}{k!},\qquad A^0=I.\]Convergence uses the operator norm \(\|A\|=\sup_{\|x\|=1}\|Ax\|\), with the Euclidean norm on vectors and \(I\) the identity matrix.
Worked example for Definition 3.11
Counterexample: Real energy eigenvalues alone do not make evolution unitary
The inequality \(\|AB\|\le\|A\|\|B\|\) follows from \(\|ABx\|\le\|A\|\|Bx\|\). Hence \(\|A^k\|\le\|A\|^k\), so the scalar exponential estimate proves convergence. Completeness here follows from entrywise completeness: \(|A_{ij}|\le\|A\|\), while \(\|A\|\le n\max_{ij}|A_{ij}|\) by Cauchy–Schwarz. Thus a Cauchy sequence of matrices has a norm limit.
Proposition 3.12. Self-adjoint matrices generate unitary evolution
Let \(H=H^*\) be self-adjoint, where \(^*\) is conjugate transpose, and let \(\hbar>0\) be a fixed real constant. By the finite spectral theorem of chapter 2, write \(H=VDV^*\), with \(V\) unitary and \(D=\operatorname{diag}(E_1,\ldots,E_n)\), \(E_j\in\mathbb R\). Then
\[U(t):=e^{-itH/\hbar}=V\operatorname{diag}(e^{-itE_1/\hbar},\ldots,e^{-itE_n/\hbar})V^*.\]For real \(s,t\), \(U(0)=I\), \(U(t+s)=U(t)U(s)\), and \(U(t)^*=U(t)^{-1}=U(-t)\). The function \(t\mapsto U(t)\) is continuous and differentiable in operator norm, and \(U'(t)=-(i/\hbar)HU(t)\).
Worked example for Proposition 3.12
Counterexample: Real energy eigenvalues alone do not make evolution unitary
Proof.
Induction gives \(H^k=VD^kV^*\). Insert this into each partial sum, then take the norm limit to obtain the displayed formula. Every diagonal entry has modulus one. Thus the diagonal matrix times its adjoint is \(I\), proving unitarity after conjugation by \(V\). The scalar addition law proves the group law entry by entry; conjugation gives the inverse formula. Scalar continuity and differentiation give the remaining claims entry by entry, hence in operator norm by the finite-dimensional bound above.
□A family with the displayed group law, unitary members, and continuous \(t\mapsto U(t)x\) for each vector \(x\) is called a strongly continuous one-parameter unitary group. Our finite-dimensional family satisfies this because \(\|(U(t)-U(s))x\|\le\|U(t)-U(s)\|\|x\|\). For \(\psi(t)=U(t)\psi_0\), the derivative gives
\[i\hbar\psi'(t)=H\psi(t),\qquad \|\psi(t)\|=\|\psi_0\|.\]Unitarity proves norm preservation. The solution is unique: differentiate \(U(-t)\psi(t)\) for any solution; the product rule and \(HU(-t)=U(-t)H\), visible in the diagonal formula, give zero. Each real component is therefore constant, fixing the solution from its initial value. The quantum framework’s autonomous dynamics postulate adopts this equation and its energy interpretation as physics; our proof establishes the matrix consequences of that adoption.
Example. Take \(H=\operatorname{diag}(0,E)\), with real \(E\ne0\), and \(\psi_0=(1,1)/\sqrt2\). Then \(\psi(t)=(1,e^{-itE/\hbar})/\sqrt2\), and its squared overlap with \(\psi_0\) is \((1+\cos(tE/\hbar))/2\). The component probabilities in this basis remain \(1/2\), but another overlap changes. An eigenvector with \(H\psi_0=E\psi_0\) instead acquires only the common phase \(e^{-itE/\hbar}\).
Counterexamples. Self-adjointness matters: on \(\mathbb C\), taking \(H=i\) gives \(e^{-itH}=e^t\), which is not unitary for \(t\ne0\). Also the scalar identity \(e^{A+B}=e^Ae^B\) is unsafe for arbitrary matrices. Let \(A=\begin{pmatrix}0&1\\0&0\end{pmatrix}\) and \(B=\begin{pmatrix}0&0\\1&0\end{pmatrix}\). Then \(e^Ae^B=(I+A)(I+B)=\begin{pmatrix}2&1\\1&1\end{pmatrix}\), whereas \((A+B)^2=I\), so the two diagonal entries of \(e^{A+B}\) are equal. The matrices differ. If \(AB=BA\), the binomial formula and the same absolute double-tail estimate as Proposition 3.6 prove the addition identity. In particular multiples of a single matrix commute.
The three expressions \(e^z\), \(e^A\), and \(e^{-itH/\hbar}\) for an unbounded operator require different definitions. The first is a scalar series; the second is a convergent matrix series. An unbounded operator need not act on every vector, and its powers need not share a domain, so substituting it into that series is not a definition on the whole Hilbert space. The live autonomous dynamics page defines its evolution through the spectral calculus. Chapter 6 supplies that vocabulary and the domain restriction on the Schrödinger derivative. The finite matrix proof does not remove that restriction.
3.6. A small Fourier dictionary
The framework’s position and momentum operators also use complex exponentials of real position variables. A plane wave is the function \(x\mapsto e^{ik\cdot x}\) on \(\mathbb R^d\), where \(k\cdot x=\sum_{j=1}^d k_jx_j\) and each \(k_j\) is real. It has modulus one and \(\partial_j e^{ik\cdot x}=ik_je^{ik\cdot x}\), by the scalar derivative formula. It is a useful function for calculation; on the entire real line its squared modulus has infinite integral because \(\int_{-R}^R1\,dx=2R\).
For a continuous compactly supported \(f:\mathbb R\to\mathbb C\), meaning \(f\) vanishes outside some bounded interval, define one convention for its Fourier transform by
\[\widehat f(k)=\int_{\mathbb R}f(x)e^{-ikx}\,dx.\]This integral is an ordinary interval integral over any interval containing the support. This is the unnormalized convention in Alphabeta’s Fourier definition. For example, if \(f\) is the indicator of \([-a,a]\), meaning one on that interval and zero elsewhere, with \(a>0\), the same formula is well-defined as a piecewise continuous Riemann integral and gives
\[\widehat f(k)=\begin{cases}2\sin(ka)/k,&k\ne0,\\2a,&k=0.\end{cases}\]Indeed \(e^{-ikx}/(-ik)\) is a primitive for \(k\ne0\), and Euler’s formula evaluates its endpoint difference. At zero the integrand is one; continuity of the result follows from \(\sin t/t\to1\), the derivative of sine at zero.
This convention names a transform and proves a simple example. It supplies no inverse formula or norm identity. Normalization factors, the conversion \(p=\hbar k\), and extension to square-integrable functions must be stated when used. In particular, a theorem such as Plancherel’s theorem, which extends a suitably normalized Fourier transform as a norm-preserving map on a completed function space, is an additional result in real and functional analysis; it does not follow from the definition above. The opening phase and matrix calculations need none of those further results.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. Complex arithmetic as plane geometry
Applies to: Definition 3.1, Proposition 3.2.
For \(z=3+4i,w=1-i\), \(\bar z=3-4i\), \(|z|=5\), \(z\bar z=25\), and \(z^{-1}=(3-4i)/25\). Their product is \(7+i\), with modulus \(\sqrt{50}=5\sqrt2=|z||w|\); its conjugate is \(7-i=\bar z\bar w\). Their sum is \(4+3i\) of modulus 5, at most \(5+\sqrt2\). These calculations check the conjugation, inverse, product, and triangle rules. The distance from \(z\) to \(w\) is \(|2+5i|=\sqrt{29}\), the Euclidean distance between their coordinate pairs.
Worked example 2. Real time derivatives and a complex integral
Applies to: Definition 3.3, Definition 3.4.
For real \(t\), \(f(t)=t^2+it\) is smooth componentwise. Its real-increment difference quotient is \(2t+h+i\to2t+i\). On \([0,1]\), \(\int f=1/3+i/2\), by the two real integrals. The vector function \((f(t),\bar f(t))\) consequently has integral \((1/3+i/2,1/3-i/2)\). Reversing endpoints negates each component. With two real coordinates, \(g(x,y)=x^2+iy\) has continuous partials \(g_x=2x,g_y=i\), illustrating the real-variable \(C^1\) definition independently of holomorphicity.
Worked example 3. An exponential on the unit circle
Applies to: Definition 3.5, Proposition 3.6, Definition 3.7, Proposition 3.8.
For \(z=it\), the absolute-value series is \(\sum |t|^n/n!\); on \(|t|\le1\), the tail after order \(N\) is at most \(2/(N+1)!\), since successive term ratios are at most \(1/2\). Thus the defining series converges uniformly there and can be separated into its real even and imaginary odd terms, giving \(\cos t+i\sin t\). At \(t=\pi/2\) it equals \(i\), so squaring and the addition law give \(e^{i\pi}=-1\). Conjugation gives \(e^{-it}\); multiplication yields \(|e^{it}|^2=1\). Differentiating gives \(ie^{it}=-\sin t+i\cos t\), explicitly checking the circle velocity. For a non-unit-modulus case, \(e^{1+i\pi/2}=ei\), of modulus \(e\); its inverse is \(-i/e\), as the exponential inverse law requires.
Worked example 4. A holomorphic polynomial and a failed direction test
Applies to: Definition 3.9, Proposition 3.10.
On the whole complex plane take \(f(z)=z^2\). The quotient \(((a+h)^2-a^2)/h=2a+h\) tends to \(2a\) in every complex direction, so \(f\) is entire. Writing \(u=x^2-y^2,v=2xy\) gives \(u_x=2x=v_y\), \(u_y=-2y=-v_x\), with continuous partials. Thus the Cauchy–Riemann criterion gives the same derivative \(2x+2iy\). For \(g(z)=\bar z\), the quotient \(\bar h/h\) is 1 for real \(h\) and \(-1\) for imaginary \(h\). It has no complex derivative; equivalently \(u_x=1\ne-1=v_y\). This explicitly illustrates both the hypotheses of the converse and why a real parameter derivative is a different condition.
Worked example 5. Matrix evolution and a Fourier integral
Applies to: Definition 3.11, Proposition 3.12.
For \(H=\begin{pmatrix}0&1\\1&0\end{pmatrix}\), \(H^*=H,H^2=I\), and \(\hbar=1\), grouping the norm-convergent matrix series gives \(U(t)=e^{-itH}=\cos t\,I-i\sin t\,H\). Multiplying by its adjoint gives \(I\), and trigonometric addition gives \(U(t+s)=U(t)U(s)\). Its derivative is \(-iHU(t)\); thus \(U(t)e_1=(\cos t,-i\sin t)\) solves \(i\psi'=H\psi\) and has norm one. If another solution has the same initial value, differentiating \(U(-t)\psi(t)\) gives zero, proving uniqueness. All vectors are in the matrix domain.
For the unboxed Fourier definition take the continuous compactly supported triangular function \(f(x)=(1-|x|)\) on \([-1,1]\), zero outside. Its transform is \(2\int_0^1(1-x)\cos kx\,dx=2(1-\cos k)/k^2\) for \(k\ne0\), by integration by parts, and equals 1 at zero. The latter value also follows from the limit \((1-\cos k)/k^2\to1/2\). A plane wave \(e^{2ix}\) instead has derivative \(2ie^{2ix}\) and squared modulus one, hence infinite squared integral on \(\mathbb R\). This checks the Fourier convention and the failure of a whole-line plane wave to be a normalized state.
Further proof details
The Fourier integral’s integration-by-parts step can be written explicitly. For \(k\ne0\), a primitive of \((1-x)\cos kx\) is \((1-x)\sin(kx)/k-\cos(kx)/k^2\). Its values at 1 and 0 differ by \((1-\cos k)/k^2\); evenness cancels the imaginary sine integral and doubles this value. At zero, the triangle’s area is \(2\int_0^1(1-x)dx=1\). Also \(2(1-\cos k)/k^2=(\sin(k/2)/(k/2))^2\to1\), using the sine derivative at zero proved from the exponential in this chapter. Thus the two branches fit continuously, with the declared Fourier convention and no appeal to an inverse-transform theorem.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. Real energy eigenvalues alone do not make evolution unitary
Related statements: Definition 3.11, Proposition 3.12.
Claim refuted. If all eigenvalues of a matrix H are real, exp(−itH) preserves norms.
Proof.
Let \(H=\begin{pmatrix}0&1\\0&0\end{pmatrix}\), with domain all of \(\mathbb C^2\), and take \(\hbar=1\). It has only eigenvalue zero: \(H(x,y)=(y,0)=\lambda(x,y)\) forces \(\lambda=0\) for any nonzero eigenvector. But \(H^2=0\), so the exponential series terminates: \(U(t)=I-itH\). It obeys \(U(t+s)=U(t)U(s)\), \(U(0)=I\), and \(iU'(t)=HU(t)\), all by multiplying this two-term expression. Nevertheless \(U(t)e_2=(-it,1)\) has norm \(\sqrt{1+t^2}\), not 1 when \(t\ne0\). The missing hypothesis in Proposition 3.12 is \(H=H^*\), not merely a real spectrum. The existing noncommuting-exponential and Cauchy–Riemann counterexamples remain valid for their separate claims.
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