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--- schrodinger-equation [May 23, 2026 at 00:03] – created Ivan Janevski
+++ schrodinger-equation [June 13, 2026 at 03:13] (current) – external edit 127.0.0.1
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 # Schrödinger equation
-In quantum mechanics, *wave function* is a mathematical object used to  describe a quantum particle. This wave function evolves according to the Schrödinger equation. For a single particle, the time-dependent 3-dimensional Schrödinger equation is stated as the following:
+The **Schrödinger equation** is the equation of motion for a quantum state. In the qubit picture the state is a complex vector $\lvert\psi\rangle$ in a Hilbert space, and the equation is a first-order linear ODE on that vector. The generator of the evolution is a Hermitian operator $H$ called the **Hamiltonian**, which encodes the total energy of the system.
-$$ i\hbar \frac{\partial}{\partial t}\Psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right) \Psi$$
+$$i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\lvert\psi\rangle = H\lvert\psi\rangle$$
-Meaning of symbols are:
+## Derivation from unitary evolution
-  * $i$ - Imaginary unit (A solution to the equation $i^2 + 1 = 0$)
-  * $\hbar$ - Reduced planck constant ($\hbar = h/2\pi$, with value $\hbar \approx 1.054 571 817\times 10^{-34}J\cdot s$)
-  * $\partial / \partial t$ - Derivative with respect to time
-  * $\nabla^2$ - Sum of second derivatives with respect to space ($\nabla^2 = \partial^2 / \partial x^2 + \partial^2 / \partial y^2 + \partial^2 / \partial z^2$)
-  * $\Psi$ - Particle wave function
-  * $m$ - Mass of the particle
-  * $V$ - Potential energy of the particle
-Because quantum theory is linear, multiple particles combine their wave functions in a linear fashion (i.e. they just add up) and form a single unified wave function that describes the entire system. This combined wave function also evolves according to Schrödinger equation.
+The central requirement on quantum time evolution is that it preserves the norm of $\lvert\psi\rangle$ — probabilities must always sum to 1. Any norm-preserving linear map is unitary, so the state at time $t + dt$ must be related to the state at time $t$ by a unitary operator:
-Note that Schrödinger equation is single-order with respect to time, but second-order with respect to space. This is unlike the classical wave equation describing (the equation describing sound and water waves), which is second-order both in time and space.
+$$\lvert\psi(t+dt)\rangle = U(t+dt,\,t)\,\lvert\psi(t)\rangle$$
-## Derivation
+For infinitesimal $dt$, continuity forces $U(t,t) = I$, so the operator can only differ from the identity by a first-order term. Writing $U(t+dt, t) = I - \frac{i}{\hbar}H\,dt$, the unitarity condition $U^\dagger U = I$ gives:
-We start from the well-known equation of energy conservation:
-$$ E = E_K + V$$
-Where $E$ is the total energy of a particle, $E_K$ is its kinetic energy and $V$ is its potential energy. **Kinetic energy** of a particle is defined as $E_K = mv^2 / 2$ while **momentum** of a particle is defined as $p = mv$. Solving the latter for $v$ and substituting it in $E_K$, then substituting $E_K$ into the equation for total energy $E$, we get the following equation:
-$$E = \frac{p^2}{2m} + V$$
-We promote energy $E$ into an operator $\hat{E} = i\hbar \partial / \partial t$ and similarily we promote $p$ the momentum into an operator $\hat{p} = -i\hbar\nabla^2$:
-$$\hat{E}\Psi = - \frac{\hat{p}^2}{2m}\Psi + V\Psi$$
-$$\boxed{i\hbar \frac{\partial\Psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right)\Psi}$$
-Thus, we directly derived the Schrödinger simply by promoting $E$ and $p$ to operators. But of course, this derivation can only be understood as informal.
-## Limitations (relativity)
+$$\left(I + \tfrac{i}{\hbar}H^\dagger\,dt\right)\!\left(I - \tfrac{i}{\hbar}H\,dt\right) = I + \tfrac{i}{\hbar}(H^\dagger - H)\,dt + O(dt^2) = I$$
-We derived the Schrödinger from nonrelativistic energy-momentum relation $E = \frac{p^2}{2m} + V$ which makes the Schrödinger inherently nonrelativistic i.e. it cannot describe particles moving close to the speed of light nor particles with no mass (which always travel at the speed of light). We would have to have used the relativistic energy-momentum relation instead:
-$$E^2 = p^2c^2 + m^2c^4 $$
-Promoting $E$ and $p$ to operators $\hat{E} = i\hbar \partial / \partial t$ and $\hat{p} = -i\hbar\nabla^2$ yields the Klein-Gordon equation
-$$\left(\frac{\partial^2\Psi}{\partial t^2} - c^2\nabla^2 + \frac{m^2c^4}{\hbar^2}\right)\Psi = 0$$
-The issue with this equation is that, unlike Schrödinger, Klein-Gordon equation is a differential equation that is second-order in time.
-It was Paul Dirac who correctly found a relativistic analogue of Schrödinger equation by postulating the existence of $\alpha, \beta, \gamma, \delta$ such that:
+which requires $H^\dagger = H$ — the generator must be Hermitian. Rearranging the finite-difference definition of the derivative then yields the Schrödinger equation.
-$$E = \sqrt{p^2c^2 + m^2c^4} = \alpha p_x c + \beta p_y c + \gamma p_z c + \delta mc^2$$
-As it turns out, $\alpha, \beta, \gamma, \delta$ cannot be ordinary complex numbers - they must be 4x4 complex matrices. Thus, wavefunction $\Psi$ in Dirac's equation is no longer a complex wave function but a 4-component Dirac spinor.
-Still, Dirac's equation does not describe creation and anahilation of particles which we see e.g. electron-positron interaction. The theory further refined by promoting wave function itself to an operator acting on Fock space (combining spaces containing 0 particles, 1 particle, 2 particles, ...).
+## Time evolution operator
+For a time-independent Hamiltonian, the Schrödinger equation has an exact closed-form solution:
+$$\lvert\psi(t)\rangle = U(t)\,\lvert\psi(0)\rangle, \qquad U(t) = e^{-iHt/\hbar}$$
+The matrix exponential is defined by its power series $e^A = \sum_{k=0}^\infty A^k / k!$. Because $H$ is Hermitian, $-iHt/\hbar$ is skew-Hermitian (satisfies $A^\dagger = -A$), and the exponential of any skew-Hermitian matrix is unitary — so $U(t)$ is guaranteed to be unitary for all $t$, consistent with the derivation above.
+## Lie group and Lie algebra
+The family of operators $\{U(t) : t \in \RR\}$ forms a one-parameter subgroup of the Lie group $\mathrm{U}(N)$ of $N\times N$ unitary matrices. The Hamiltonian $-iH$ (or $-iH/\hbar$ in SI units) belongs to the corresponding Lie algebra $\mathfrak{u}(N)$ of skew-Hermitian matrices. The exponential map $\exp : \mathfrak{u}(N) \to \mathrm{U}(N)$ is exactly the matrix exponential above — it converts an element of the algebra (a Hamiltonian) into an element of the group (a gate).
+This correspondence means that specifying a Hermitian operator $H$ is equivalent to specifying a continuous family of unitary transformations $e^{-iHt}$ parametrized by time (or by a rotation angle). Every single-qubit and multi-qubit gate can be written in this form for some $H$.
+## Quantum gates
+When the Hamiltonian satisfies $H^2 = I$ — which holds for all Pauli matrices $X$, $Y$, $Z$ — the power series collapses to a two-term identity via Euler's formula:
+$$e^{-i\theta H} = \cos\theta\,I - i\sin\theta\,H$$
+The single-qubit rotation gates are defined exactly this way:
+$$R_x(\theta) = e^{-i\theta X/2}, \quad R_y(\theta) = e^{-i\theta Y/2}, \quad R_z(\theta) = e^{-i\theta Z/2}$$
+The factor of $1/2$ means a full rotation of the Bloch sphere requires $\theta = 2\pi$ rather than $\pi$, which is why a $180°$ physical rotation maps to a $\pi$ gate argument. The [[h-gate|Hadamard gate]] can be written as $H = e^{i\pi(X+Z)/(2\sqrt{2})}$, placing it in the same framework.
+## List of code implementations
+ - [[schrodinger-equation-qutip|Schrödinger equation (QuTiP)]]
-This is called the second quantization, and it's the framework of quantum field theory (QFT). Also, in QFT, Klein-Gordon equation rises from the ashes successfuly being able to describe spin-0 partcles.