Wavefunction Collapse
In traditional textbook quantum mechanics, wavefunction collapse means that when a measurement produces a definite outcome, the state is updated from a superposition of possible measurement outcomes to the state corresponding to the outcome actually obtained.
For example, suppose the state before measuring the path is
\[|\psi\rangle=\frac{|A\rangle+|B\rangle}{\sqrt2}.\]
If the measurement produces outcome A, textbook quantum mechanics represents the update as
\[\boxed{\frac{|A\rangle+|B\rangle}{\sqrt2}\longrightarrow|A\rangle}.\]
If the measurement instead produces B, the state is updated to \(|B\rangle\).
This rule tells us how to update the quantum state after obtaining a measurement result. It does not by itself provide a universally agreed physical mechanism explaining why one particular outcome occurs.
Collapse Is Associated with Measurement
In the traditional textbook formulation, collapse is associated with a measurement producing a definite result.
This creates an important question. If measurement is ultimately a physical interaction between a quantum system and a detector, why should that interaction behave differently from ordinary quantum evolution?
Under ordinary quantum dynamics, an interaction with a detector can produce
\[\frac{|A\rangle|D_A\rangle+|B\rangle|D_B\rangle}{\sqrt2}\]
rather than simply selecting one of the two terms.
This leads to the question of exactly what counts as a measurement and is part of the quantum measurement problem.
What Does Quantum Evolution Mean?
Quantum evolution simply means how a quantum state changes as time passes.
If a system initially has state
\[|\psi(0)\rangle\]
and later has state
\[|\psi(t)\rangle,\]
then
\[|\psi(0)\rangle\longrightarrow|\psi(t)\rangle\]
describes its quantum time evolution.
In classical mechanics, Newton’s laws tell us how quantities such as position and velocity change with time. In ordinary nonrelativistic quantum mechanics, the Schrödinger equation tells us how the quantum state changes with time.
Why Does Ordinary Evolution Preserve Superposition?
Suppose two states evolve according to
\[|A\rangle\longrightarrow|A’\rangle\]
and
\[|B\rangle\longrightarrow|B’\rangle.\]
If ordinary quantum evolution is represented by a linear operator \(U\), then
\[U(\alpha|A\rangle+\beta|B\rangle)=\alpha U|A\rangle+\beta U|B\rangle.\]
Therefore
\[\boxed{\alpha|A\rangle+\beta|B\rangle\longrightarrow\alpha|A’\rangle+\beta|B’\rangle}.\]
The superposition structure is preserved. Ordinary linear evolution does not arbitrarily discard one component and retain the other.
Why Is the Evolution Linear?
Within standard quantum mechanics, this linearity appears directly in the Schrödinger equation:
\[\boxed{i\hbar\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle}.\]
If \(|A\rangle\) and \(|B\rangle\) are solutions, then the linear mathematical structure allows a linear combination of them to remain a solution.
For
\[|\psi\rangle=\alpha|A\rangle+\beta|B\rangle,\]
the derivative is linear:
\[\frac{d}{dt}(\alpha|A\rangle+\beta|B\rangle)=\alpha\frac{d|A\rangle}{dt}+\beta\frac{d|B\rangle}{dt}.\]
The Hamiltonian \(H\) is also a linear operator:
\[H(\alpha|A\rangle+\beta|B\rangle)=\alpha H|A\rangle+\beta H|B\rangle.\]
But This Raises a Deeper Question
Saying that \(|A\rangle\) and \(|B\rangle\) individually obey the Schrödinger equation is not yet a satisfying explanation if we have not understood the Schrödinger equation itself.
The more fundamental question is therefore:
What is the Schrödinger equation actually saying about a quantum system?
What Problem Does the Schrödinger Equation Solve?
Start with classical mechanics. If we know the position and velocity of a ball and the forces acting upon it, Newton’s laws allow us to predict how its motion changes with time.
Quantum mechanics needs an analogous dynamical law.
A quantum system is described by its quantum state:
\[|\psi\rangle.\]
Therefore quantum mechanics needs a rule answering:
If we know the quantum state right now, what will its quantum state be a moment later?
The Schrödinger equation provides that rule:
\[\boxed{i\hbar\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle}.\]
A First Conceptual Reading of the Equation
We do not yet need to understand every symbol. At this stage, the equation can be read conceptually as
\[\boxed{\text{how the quantum state is changing}\;\longleftrightarrow\;\text{the Hamiltonian acting on the state}}.\]
The symbol \(H\) is called the Hamiltonian. Roughly speaking, it represents the energy structure and interactions governing the system.
The conceptual picture is therefore
\[\boxed{\text{current quantum state}\xrightarrow[\text{Hamiltonian}]{\text{Schrödinger equation}}\text{future quantum state}}.\]
Where the Learning Should Continue
Before using the Schrödinger equation to justify further claims about quantum evolution, the next step is to understand the Hamiltonian \(H\): what it represents, how it is constructed in simple systems, and why energy is connected to the rate at which a quantum state changes.
Note metadata
- Note type: learning-note
- Subject: learning-notes
- Source: Quantum Computing for Everyone