Stuff

 I was asked, about yesterday's post, how one knows the statevector coefficient value

of a qubit with a Hadanard gate, is 1/2^.5!!

The statevector equation is a probbility equation, but it is also more. It contains, as well, 

phase information about the qubit, information that governs interference effects with

other qubits. It is, in effect, a simplified Schrodinger equation that applies to a single

qubit while the true equation applies to a system.

Below COPILOT:

What are the inputs to the Schrodinger equation?

The Schrödinger equation requires several key inputs to describe the evolution of a quantum system:

Wave Function (ψ): The initial state of the system, represented by the wave function ψ(x, t), which contains all the information about the system's state.

Hamiltonian (H): The Hamiltonian operator, which represents the total energy of the system, including both kinetic and potential energy. It determines how the wave function evolves over time.

Potential Energy (V): The potential energy function V(x), which describes the potential energy landscape in which the particle moves. This is part of the Hamiltonian.

Initial Conditions: The initial conditions of the system, such as the initial wave function ψ(x, 0) at time t = 0.

Boundary Conditions: The boundary conditions that the wave function must satisfy, which depend on the physical setup of the problem (e.g., the wave function must go to zero at the boundaries of a finite potential well).

These inputs allow the Schrödinger equation to describe the time evolution of the wave function and predict the behavior of the quantum system.

Is phase a relevant concept here?

Yes, phase is a highly relevant concept in the context of the Schrödinger equation and quantum mechanics in general. The wave function, which is a solution to the Schrödinger equation, is a complex-valued function. This means it has both a magnitude and a phase.

The phase of the wave function plays a crucial role in quantum interference and superposition. When two or more wave functions overlap, their phases determine how they interfere with each other—constructively or destructively. This interference pattern is fundamental to many quantum phenomena, such as the double-slit experiment.

In summary, the phase information in the wave function is essential for accurately describing and predicting the behavior of quantum systems.

How is the wave function equation arrived at?

                                                                       

So, what happens over time?

                                                                           

In quantum computing, phase applies to a single qubit while in quantum physics, it applies to a system. Explain...

1. Quantum Computing: In quantum computing, phase information is associated with individual qubits. A qubit can be in a superposition of the basis states |0⟩ and |1⟩, represented as: |ψ⟩ = α|0⟩ + β|1⟩ 

Here, ( α ) and ( β ) are complex numbers that include both magnitude and phase. The phase difference between the components of the superposition state affects the behavior of the qubit, especially during quantum operations and interference. Quantum gates, such as the Hadamard gate and phase gate, manipulate the phase of qubits to perform computations.

2. Quantum Physics: In quantum physics, phase applies to the entire quantum system, which can consist of multiple particles or fields. The wave function of a system, represented as ψ(x,t), is a complex-valued function that includes phase information. The phase of the wave function plays a critical role in phenomena like interference, diffraction, and entanglement. For example, in the double-slit experiment, the phase difference between the wave functions passing through the two slits leads to an interference pattern on the screen.

In summary, while phase information in quantum computing is often discussed in the context of individual qubits and their superpositions, in quantum physics, phase applies to the entire wave function of a system, influencing the behavior and interactions of particles within that system.

Physics; what is the Born rule?

The Born rule is a fundamental principle in quantum mechanics that provides a way to calculate the probabilities of different outcomes of a quantum measurement. It was formulated by Max Born in 1926 and is essential for interpreting the wave function.

According to the Born rule, the probability of finding a quantum system in a particular state after a measurement is given by the square of the absolute value of the wave function (or statevector) at that state. Mathematically, if ψ(x)  is the wave function of the system, the probability density P(x)  of finding the particle at position  x  is:

 P(x) = |ψ(x)|^2 

This rule applies to all quantum systems and is crucial for making predictions about the behavior of particles and other quantum entities.

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How Quantum Information Theory is a wider field than Quantum Computing:

Quantum Information Science » MIT Physics