Quantum Superposition - Sarcasm Wiki

Contents

1. Overview
2. Etymology
3. Cultural Impact

Ah, you want me to explain the principle of quantum mechanics regarding superposition. Don’t expect me to hold your hand. This is… necessary, I suppose. Just try to keep up.

Principle of quantum mechanics

For a more comprehensive exploration of this subject, consult the article on the Superposition principle .

Quantum superposition of states and decoherence

This section is a placeholder, really. It’s a part of a larger series on Quantum mechanics .

Here’s a glimpse of the foundational equation, the Schrödinger equation , represented in Dirac notation :

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

And the surrounding context, if you must know:

Background

Fundamentals

Experiments

Formulations

Equations

Interpretations

Advanced topics

Scientists

Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.

Take a qubit , the basic unit in quantum information processing . It’s not just a 0 or a 1. It’s a blend, a superposition of the basic states, $|0\rangle$ and $|1\rangle$. Think of it like this:

$|\Psi \rangle =c_{0}|0\rangle +c_{1}|1\rangle $

Here, $|\Psi \rangle$ is the quantum state. $|0\rangle$ and $|1\rangle$ are those specific solutions to the Schrödinger equation, weighted by these complex numbers, $c_0$ and $c_1$. These are called probability amplitudes . $|0\rangle$ is like your classical 0 bit, and $|1\rangle$ is the classical 1 bit. Now, when you actually look at it, when you measure it, the probability of finding it in the $|0\rangle$ state is $|c_0|^2$, and in the $|1\rangle$ state, it’s $|c_1|^2$. This is the Born rule , in case you were wondering. Before you measure, it’s genuinely in both states at once. It’s a mess of possibilities.

Remember those interference patterns in the double-slit experiment ? That’s superposition in action. It’s not just one path or the other; it’s a combination of both.

Wave postulate

The entire edifice of quantum mechanics is built on the postulate that a wave equation describes the state of a quantum system. And this isn’t just any equation; it’s linear and homogeneous. What does that mean for us? It means if you have two solutions, $\Psi_1$ and $\Psi_2$, you can mix them together – linearly, of course – and the resulting combination, $\Psi = c_1\Psi_1 + c_2\Psi_2$, is also a valid solution. This applies no matter how many solutions you start with. It’s a fundamental property of linearity.

Transformation

These quantum wave equations can be solved in different ways. You can describe a system using its position, $\Psi(\vec{r})$, or its momentum, $\Phi(\vec{p})$. And just like with position, if you have momentum solutions, you can combine them:

$\Phi(\vec{p}) = d_1\Phi_1(\vec{p}) + d_2\Phi_2(\vec{p})$

The link between these position and momentum descriptions? A linear transformation , specifically a Fourier transformation . This transformation itself is a kind of quantum superposition. Every position wave function can be seen as a superposition of momentum wave functions, and vice versa. These aren’t simple sums; they often involve an infinite number of component waves.

Generalization to basis states

You can also express a quantum solution as a superposition of eigenvectors . Think of these eigenvectors as representing the possible outcomes of a measurement. For an operator $\hat{A}$, an eigenvector $\psi_i$ satisfies:

$\hat{A}\psi_i = \lambda_i\psi_i$

Here, $\lambda_i$ is one of the possible values you could measure for the observable $A$. Any solution $\Psi$ can then be written as a combination of these basis states:

$\Psi = \sum_{n} a_i\psi_i$

These $\psi_i$ states? They’re the basis states.

Compact notation for superpositions

To make things less cumbersome, especially when dealing with systems that don’t have a straightforward classical analog, like quantum spin, we use Dirac notation . This allows us to work with just the coefficients of the superposition, simplifying complex calculations:

$|v\rangle = d_1|1\rangle + d_2|2\rangle$

This shorthand is ubiquitous in quantum mechanics. Superposition isn’t just a concept; it’s a workhorse.

Consequences

Paul Dirac , bless his meticulous soul, described the superposition principle quite aptly:

“The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.”

Anton Zeilinger , reflecting on the famous double-slit experiment , shed light on how superposition is both created and destroyed:

“[T]he superposition of amplitudes … is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘out there.’ The absence of any such information is the essential criterion for quantum interference to appear.”

Theory

General formalism

Any quantum state can be expressed as a sum, or superposition, of the eigenstates of an Hermitian operator. Take the Hamiltonian, for instance. Its eigenstates form a complete basis, meaning you can expand any state $|\alpha\rangle$ in terms of them:

$|\alpha\rangle =\sum {n}c{n}|n\rangle $

where $|n\rangle$ represents the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, $|x\rangle$:

$|\alpha \rangle =\int dx’|x’\rangle \langle x’|\alpha \rangle $

Here, $\phi_{\alpha}(x) = \langle x|\alpha \rangle$ is the wave function, the projection of the state onto the position basis. Both cases demonstrate that $|\alpha\rangle$ can be a superposition of an infinite number of basis states.

Example

Consider the Schrödinger equation with the Hamiltonian $\hat{H}$:

$\hat{H}|n\rangle = E_{n}|n\rangle $

$\hat{H}{\big (}|n\rangle +|n’\rangle {\big )}=E_{n}|n\rangle +E_{n’}|n’\rangle $

However, $|\Psi\rangle$ is not generally an eigenstate unless $E_n = E_{n’}$. This means $|\Psi\rangle$ is a superposition of energy eigenstates.

Let’s get more concrete. Imagine an electron with spin . We can describe its state using spin-up $|{\uparrow }\rangle$ and spin-down $|{\downarrow }\rangle$ states, indexed in the $\hat{z}$ basis:

$|\Psi \rangle =c_{1}|{\uparrow }\rangle +c_{2}|{\downarrow }\rangle $

The squares of the magnitudes of these complex coefficients, $|c_1|^2$ and $|c_2|^2$, give the probabilities of finding the electron with spin up or spin down, respectively. This is due to the Born rule . The total probability must sum to 1:

$P(|{\uparrow }\rangle )+|P(|{\downarrow }\rangle )=|c_{1}|^{2}+|c_{2}|^{2}=1$

These coefficients, $c_1$ and $c_2$, are complex. So, a state like:

$|\Psi \rangle ={\frac {3}{5}}i|{\uparrow }\rangle +{\frac {4}{5}}|{\downarrow }\rangle $

is perfectly valid. The probabilities then become:

$P(|{\uparrow }\rangle )=\left|{\frac {3i}{5}}\right|^{2}={\frac {9}{25}}$ $P(|{\downarrow }\rangle )=\left|{\frac {4}{5}}\right|^{2}={\frac {16}{25}}$

And, naturally, $\frac{9}{25} + \frac{16}{25} = 1$.

If you consider a qubit that has both position and spin properties, its state becomes a superposition of all possible combinations:

$\Psi =\psi _{+}(x)\otimes |{\uparrow }\rangle +\psi _{-}(x)\otimes |{\downarrow }\rangle $

Here, $\Psi$ is the sum of the tensor products of the position-dependent wave functions and the spin states.

Experiments

The principle of superposition isn’t just theoretical. We’ve managed to put surprisingly large objects into superposed states:

A beryllium ion has been successfully placed in a superposition state. They called it a “Schrödinger Cat” state.
The double-slit experiment , a classic demonstration of superposition, has been performed with molecules as large as buckyballs and complex organic molecules containing up to 2000 atoms.
Even larger molecules, exceeding 10,000 atomic mass units and composed of over 810 atoms, have been shown to exhibit superposition.
Highly sensitive magnetometers, known as superconducting quantum interference devices (SQUIDS), rely on quantum interference effects in superconducting circuits, which are a direct consequence of superposition.
A tiny piezoelectric “tuning fork ,” containing roughly 10 trillion atoms, has been put into a superposition of vibrating and non-vibrating states.
Intriguingly, research suggests that chlorophyll in plants might utilize quantum superposition to enhance the efficiency of energy transport, allowing pigment proteins to be spaced further apart than classical physics would predict.

In quantum computers

In quantum computers , the qubit is the fundamental unit of information, analogous to the classical bit . But unlike a bit, which is either 0 or 1, a qubit can exist in a superposition of both states. Controlling these superpositions is key to quantum computation, yet it’s also a significant challenge. The superposition needs to be stable enough to avoid unintended interactions, but interactions are precisely what we need to perform computations and read out results. Systems like nuclear spins , with their weak coupling, are robust against external noise but make readout difficult.

There. That’s the essence of superposition. It’s about possibilities coexisting until forced to choose. Don’t expect me to elaborate further unless you have a truly compelling reason.