DAY 5: My learning #Quantum30 Challenge
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| Photo by Fractal Hassan on Unsplash |
Introduction: Quantum mechanics is a branch of physics that describes the behavior of particles on a microscopic scale. One of its most intriguing phenomena is quantum superposition, which implies that quantum objects can exist in multiple states simultaneously. In this report, we will explore this concept using a two-dimensional vector space and the bra-ket notation.
The Basis Vector Representation: In quantum mechanics, we often represent quantum states as vectors in a vector space. These vectors are usually chosen to form a basis for the space. For our example, we will use a two-dimensional vector space with two basis vectors: |R⟩ for red and |G⟩ for green.
Quantum Superposition: Quantum superposition means that a quantum object can exist in a linear combination of its basis states. In our example, this can be represented as:
|ψ⟩ = α|R⟩ + β|G⟩
Here, |ψ⟩ represents the quantum state of our colored ball, and α and β are complex numbers representing the probability amplitudes of the ball being in the red and green states, respectively. The probabilities of finding the ball in the red or green state are given by |α|^2 and |β|^2, respectively.
Mathematical Example: Let's consider a mathematical example to illustrate quantum superposition using the bra-ket notation. Consider a real ball which can stay only in two colors, either red or green. Now, for a quantum ball lets say it exist in superposition of both the color, which can be represented vectorially like this.
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| fig.01 |
Why length of 1 unit ?Note that the length of vector is always kept 1, unlike a normal real life velocity vector. The fact it is assigned a value one is to measure the probability. If we know a object is completely red, then the quantum objects wave function is simply:|ψ⟩ = 1|R⟩ + 0|G⟩So, on squaring this always result in 1 or 100% of chance only lying in red colored state
|ψ⟩ = (0.8704|R⟩ + (0.4923i)|G⟩
Now, we want to calculate the probability of finding the ball in the red state when measured. To do this, we use the bra-ket notation and the concept of the inner product:
P(Red) = |⟨R|ψ⟩|^2
Using the given values:
P(Red) = |⟨R|((0.8704)|R⟩ + (0.4923i)|G⟩)|^2
P(Red) = |(0.8704)|R⟩ + (0.4923i)|G⟩)|^2
P(Red) = (0.8704)|R⟩ + (0.4923i)|G⟩) * (0.8704)|R⟩ - (0.4923i)|G⟩)
Simplifying:
P(Red) = (0.8704)(0.8704)|⟨R|R⟩ + (0.4923i)(-0.4923i)|⟨G|R⟩
P(Red) = (0.8704)*(0.8704)*1 + 0.4923*0.4923i*0
P(Red) = 75.7640%
So, the probability of finding the ball in the red state is 75.7640%.
Now, let's represent this concept using a 3-dimensional bra-ket notation: In quantum mechanics, we often use a ket notation to represent quantum states. For a three-dimensional vector space, we can have three basis vectors, typically denoted as |x⟩, |y⟩, and |z⟩. These basis vectors represent the possible states of a quantum object along the x, y, and z axes, respectively.
A quantum object can exist in a superposition of these states, which we can represent as:
|ψ⟩ = α|x⟩ + β|y⟩ + γ|z⟩
Here, |ψ⟩ represents the quantum state of the object, and α, β, and γ are complex coefficients that determine the probability amplitudes of the object being in the states |x⟩, |y⟩, and |z⟩, respectively. When we measure the object's state, it will collapse into one of these states with probabilities given by |α|^2, |β|^2, and |γ|^2.
Uncertainty and Superposition: In the realm of quantum mechanics, a fundamental principle known as the Heisenberg Uncertainty Principle dictates that both the precise position and wavelength (momentum) of a quantum object cannot be simultaneously measured with absolute certainty. Attempting to measure one of these properties with great precision inherently disrupts the accuracy of our knowledge regarding the other. Consequently, quantum objects persist in a state of superposition, embodying a multitude of possible basic vectors representing their position and momentum.
Upon conducting a measurement, the quantum object's superposition collapses, and a specific outcome emerges in the physical environment. This outcome is inherently probabilistic, with the likelihood of various results governed by the complex interplay of the object's superposition coefficients. Thus, the very act of measurement not only reveals a quantum object's property but also plays a role in determining the outcome itself, highlighting the intriguing and often counterintuitive nature of quantum mechanics.
Conclusion: In this report, we introduced the concept of quantum superposition using the bra-ket notation and a simple example involving red and green colored balls. Quantum superposition allows quantum objects to exist in multiple states simultaneously, and we illustrated how to calculate probabilities within this framework. Understanding quantum superposition is essential for grasping the peculiar behavior of quantum systems, and it forms the foundation of many quantum phenomena.
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