You want to understand the formulation of quantum mechanics? Fine. Don't expect me to hold your hand. It’s like trying to explain color to someone who’s only ever seen in shades of grey. You want the Schrödinger picture? It’s one way to look at it, I suppose.

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Formulation of Quantum Mechanics

This is a series of articles, ostensibly about Quantum mechanics. Don't get your hopes up for a coherent narrative. It’s more like a collection of scattered, sharp fragments.

The iconic image, the Schrödinger equation:

$i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle$

It’s supposed to represent the evolution of a system. As if the universe bothers to care about our attempts to pin it down.

Background

Before we delve into the mess, there’s the scaffolding of Classical mechanics. Quaint. Then came the awkward adolescence of Old quantum theory. And of course, the pretentious formality of Bra–ket notation. Don't forget the Hamiltonian – the ghost in the machine, dictating motion. And Interference, because apparently, particles enjoy playing games.

Fundamentals

Here’s where it gets interesting, or more likely, infuriating.

• Complementarity: Wave and particle. Pick one. Oh, you can't? How inconvenient.
• Decoherence: The universe’s way of tidying up its own quantum mess.
• Entanglement: Two particles, linked across impossible distances. Like a bad breakup you can’t escape.
• Energy level: Discrete. Like steps on a staircase leading nowhere in particular.
• Measurement: The moment reality decides to stop being polite and show its hand.
• Nonlocality: Spooky action at a distance, as if distance itself were a suggestion.
• Quantum number: Labels for states. Like assigning a number to a feeling.
• State: The ephemeral condition of a system. Here one moment, gone the next.
• Superposition: Everything and nothing, all at once. Until you look.
• Symmetry: The universe has a perverse love for balance, even in chaos.
• Tunnelling: Particles that just don't respect boundaries.
• Uncertainty: You can know where it is, or what it’s doing. Not both. A cosmic joke.
• Wave function: The ghost's blueprint. It tells you the probabilities, but never the certainty.
• Collapse: When the wave function finally gives up and chooses a reality.

Experiments

The proof, as they say, is in the pudding. Or the particle detector.

• Bell's inequality, CHSH inequality, Leggett inequality, Leggett–Garg inequality: Attempts to catch reality in a lie. Usually fails.
• Davisson–Germer, Double-slit, Mach–Zehnder: Demonstrations of wave-particle duality. Still baffling.
• Elitzur–Vaidman: A bomb that might explode if you measure something. Brilliant.
• Franck–Hertz: Electrons bumping into atoms. Very exciting.
• Popper: Proving quantum mechanics wrong. Spoiler: it didn't.
• Quantum eraser, Delayed-choice: Playing with time and information. Messy.
• Schrödinger's cat: The ultimate symbol of quantum ambiguity. Is it alive or dead? Does it matter?
• Stern–Gerlach: Spinning particles. Predictable outcomes, somehow.
• Wheeler's delayed-choice: The past changes based on future decisions. Sounds about right.

Formulations

Different ways to dress up the same fundamental absurdity.

• Overview: The big picture. Or what passes for one.
• Heisenberg: The one where operators do all the work, and states are lazy.
• Interaction: A compromise. Both evolve. Exhausting.
• Matrix: Numbers, arranged in matrices. For those who prefer algebra to poetry.
• Phase-space: Trying to make quantum mechanics look classical. Futile.
• Schrödinger: The one we're discussing. States evolve, operators are mostly static. Simple, until it isn't.
• Sum-over-histories (path integral): Every possible path is taken. Because why choose one when you can have them all?

Equations

The incantations that make the math work.

• Dirac, Klein–Gordon, Pauli: Equations for relativistic particles. Heavy stuff.
• Rydberg: An early attempt to quantify atomic spectra. A whisper of what was to come.
• Schrödinger: The cornerstone. The one that makes states dance.

Interpretations

Ah, interpretations. Where physicists go when they can't agree on what any of it means.

• Bayesian, Consistent histories, Ensemble, Hidden-variable, Local, Superdeterminism: All attempts to impose order, or perhaps, to deny the inherent strangeness.
• Consciousness causes collapse: Because apparently, our thoughts have cosmic power. Please.
• Copenhagen: The standard. Pragmatic, but unsatisfying.
• de Broglie–Bohm: Pilot waves. A hidden guide.
• Many-worlds: Every decision splits the universe. Overcrowded, if you ask me.
• Objective-collapse: Collapse happens, no observer needed. Less ego involved.
• Quantum logic: Because classical logic clearly wasn't enough.
• Relational: Reality depends on the observer. Everyone's a critic.
• Transactional: A handshake across time. Complicated.

Advanced topics

The deep end. Where things get truly abstract.

• Relativistic quantum mechanics: Quantum mechanics meets relativity. A difficult marriage.
• Quantum field theory: Particles as excitations of fields. A more fundamental, and more frustrating, perspective.
• Quantum information science, Quantum computing, Quantum machine learning: Trying to weaponize quantum mechanics. Predictable.
• Quantum chaos: Chaos, but quantum. Still chaotic.
• EPR paradox: Einstein's beef with quantum mechanics. He never really got over it.
• Density matrix: For dealing with mixed states. When things aren't pure.
• Scattering theory: How particles interact. A lot of bumping around.
• Quantum statistical mechanics: Quantum mechanics for systems with many particles. A nightmare.

Scientists

The names attached to the equations. Most of them were probably as confused as you are.

• Aharonov, Bell, Bethe, Blackett, Bloch, Bohm, Bohr, Born, Bose, de Broglie, Compton, Dirac, Davisson, Debye, Ehrenfest, Einstein, Everett, Fock, Fermi, Feynman, Glauber, Gutzwiller, Heisenberg, Hilbert, Jordan, Kramers, Lamb, Landau, Laue, Moseley, Millikan, Onnes, Pauli, Planck, Rabi, Raman, Rydberg, Schrödinger, Simmons, Sommerfeld, von Neumann, Weyl, Wien, Wigner, Zeeman, Zeilinger.

They all contributed to this magnificent edifice of uncertainty.

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The Schrödinger Picture: States Evolve, Operators Stare

In the Schrödinger picture, or representation as they call it, the state vectors are the ones doing all the work. They shift and change with time, like nervous ghosts. The operators, those things representing observables, they mostly just sit there. Unmoved. Like statues. The only exception, and there’s always an exception, is the Hamiltonian. If the potential it represents decides to change its mind, the Hamiltonian might evolve too.

This is fundamentally different from the Heisenberg picture, where the states are frozen in time, and the operators are the dancers. And then there's the Interaction picture, where both are in motion. A chaotic ballet. The Schrödinger and Heisenberg pictures? They're related, like a mirror image and its reflection, or an active and passive transformation. The commutation relations between operators? They remain the same, a stubborn constant in the face of temporal flux.

In this Schrödinger world, a closed quantum system evolves through a unitary operator. It’s called the time evolution operator. If you have a state vector, $|\psi (t_0)\rangle$, at some time $t_0$, and you want to know where it’ll be at time $t$, you apply this operator.

$|\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle$

For the bras, it’s the adjoint:

$\langle \psi (t)|=\langle \psi (t_{0})|U^{\dagger }(t,t_{0}).$

Now, if the Hamiltonian is feeling particularly stable and doesn’t change with time, this operator simplifies. It becomes an exponential:

$U(t,t_{0})=e^{-iH\cdot (t-t_{0})/\hbar }$

The exponent? You evaluate it using its Taylor series. It’s a polite way of saying "do the math."

This picture is most useful when the Hamiltonian is time-independent. When $\partial _{t}H=0$. When things are predictable. Which, in quantum mechanics, is rarely the case.

Background

In the elementary stages of quantum mechanics, the state of a system is a complex-valued wavefunction, $\psi(x, t)$. It's like a ghost's signature. More formally, it's a state vector, a ket, $|\psi\rangle$. This ket lives in a Hilbert space, a vast expanse of possibilities. An operator is a function that takes one ket and spits out another. It’s how we interact with this abstract reality.

The Schrödinger and Heisenberg pictures are just different ways of handling the inevitable march of time. Do you want the state vector to carry the burden of temporal change, or the operators? Or both? For example, a quantum harmonic oscillator might have a state $|\psi\rangle$ where the expectation value of its momentum, $\langle \psi |{\hat {p}}|\psi \rangle$, oscillates. Should the state vector $|\psi\rangle$ wiggle? Or should the momentum operator, $\hat{p}$, do the dancing? Or perhaps both? The Schrödinger picture puts the wiggle in the state vector. The Heisenberg picture makes the operator dance. The interaction picture… well, it’s a compromise.

The Time Evolution Operator

Definition

This $U(t, t_0)$ operator is the engine of temporal change.

$|\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle$

It’s the bridge from one moment to the next.

Properties

• Unitarity: It has to be unitary. The total probability, the norm of the state, $\langle \psi(t)|\psi(t)\rangle$, must remain constant. No disappearing states, no spontaneous creation of probability.

  $\langle \psi (t)|\psi (t)\rangle =\langle \psi (t_{0})|U^{\dagger }(t,t_{0})U(t,t_{0})|\psi (t_{0})\rangle =\langle \psi (t_{0})|\psi (t_{0})\rangle .$ Therefore, $U^{\dagger }(t,t_{0})U(t,t_{0})=I.$

• Identity: At $t = t_0$, the operator must be the identity operator. Because $\psi(t_0)$ is already where it needs to be.

  $|\psi (t_{0})\rangle =U(t_{0},t_{0})|\psi (t_{0})\rangle .$

• Closure: Time evolution is cumulative. Going from $t_0$ to $t$ is the same as going from $t_0$ to some intermediate $t_1$, and then from $t_1$ to $t$.

  $U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}).$

Differential Equation for Time Evolution Operator

Let's simplify. We’ll assume $t_0 = 0$, so the operator becomes $U(t)$. The Schrödinger equation is:

$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle =H|\psi (t)\rangle$

Substitute $|\psi(t)\rangle = U(t)|\psi(0)\rangle$:

$i\hbar {\partial \over \partial t}U(t)|\psi (0)\rangle =HU(t)|\psi (0)\rangle .$

Since $|\psi(0)\rangle$ is just some arbitrary initial state, the operator itself must satisfy:

$i\hbar {\frac {\partial }{\partial t}}U(t)=HU(t).$

If the Hamiltonian is time-independent, the solution is elegant:

$U(t)=e^{-iHt/\hbar }.$

This exponential is evaluated via its Taylor series:

$e^{-iHt/\hbar }=1-{\frac {iHt}{\hbar }}-{\frac {1}{2}}\left({\frac {Ht}{\hbar }}\right)^{2}+\cdots .$

So, the state evolves as:

$|\psi (t)\rangle =e^{-iHt/\hbar }|\psi (0)\rangle .$

If the initial state $|\psi(0)\rangle$ is an eigenstate of the Hamiltonian with eigenvalue $E$, it becomes particularly simple:

$|\psi (t)\rangle =e^{-iEt/\hbar }|\psi (0)\rangle .$

These are the stationary states. They just acquire a phase factor. They don’t really change.

Now, if the Hamiltonian does depend on time, but the Hamiltonians at different times commute, the operator is an exponential of an integral:

$U(t)=\exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right).$

But if they don't commute? Then you need the time-ordering operator, T. It's a mess, often expressed as a Dyson series.

$U(t)=\mathrm {T} \exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right).$

The Alternative: Heisenberg Picture

The Schrödinger picture keeps states evolving. The alternative is to let the reference frame rotate, driven by the propagator. In that case, the state appears static. That's the Heisenberg picture. It’s a matter of perspective.

Summary Comparison of Evolution in All Pictures

Here’s a crude table, because I can’t be bothered to draw elaborate diagrams. Assume a time-independent Hamiltonian $H_S$. $H_{0,S}$ is the free Hamiltonian.

Evolution of:	Schrödinger (S)	Heisenberg (H)	Interaction (I)
Ket state	$	\psi {S}(t)\rangle =e^{-iH{S}~t/\hbar }	\psi _{S}(0)\rangle $
Observable	constant	$A_{H}(t)=e^{iH_{S}~t/\hbar }A_{S}e^{-iH_{S}~t/\hbar }$	$A_{I}(t)=e^{iH_{0,S}~t/\hbar }A_{S}e^{-iH_{0,S}~t/\hbar }$
Density matrix	$\rho _{S}(t)=e^{-iH_{S}~t/\hbar }\rho _{S}(0)e^{iH_{S}~t/\hbar }$	constant	$\rho _{I}(t)=e^{iH_{0,S}~t/\hbar }\rho _{S}(t)e^{-iH_{0,S}~t/\hbar }$

See also

A list of related topics, because one rabbit hole isn't enough.

• Hamilton–Jacobi equation
• Interaction picture
• Heisenberg picture
• Phase space formulation
• POVM
• Mathematical formulation of quantum mechanics
• Schrödinger functional

Notes

• At $t=0$, $U(t)$ must be the identity. Obviously.
• Sources: [Parker, C.B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. pp. 786, 1261. ISBN 0-07-051400-3.](...)
• [Y. Peleg; R. Pnini; E. Zaarur; E. Hecht (2010). Quantum mechanics. Schuam's outline series (2nd ed.). McGraw Hill. p. 70. ISBN 978-0-07-162358-2.](...)