Superposition - Beyond the Misconceptions
Day 5- Quantum30 challenge
As we approach the end of the first week of
’s #Quantum30 challenge, it’s essential to explore the concept of ‘Superposition’ in quantum physics. This article provides more of a mathematical interpretation of the term.Superposition. Whenever you search for this word in the context of Quantum Mechanics, the general and most common answer you get is
‘Superposition is the ability of a (quantum) particle to exist in multiple states at the same time until its measured’
This is the definition that I got when I Googled the term. Though for a beginner, this is good and enough, this definition can be misleading at times. Let’s refine our understanding of this concept and clarify any misconceptions that might arise from preconceived notions. In yesterday’s post, we defined a wave function as a mathematical tool to represent a particle. So obviously, there is some abstract mathematics involved.
The Basis of Superposition: Linear Algebra and Vector Spaces
To comprehend superposition fully, we must first explore its mathematical foundations. In linear algebra, we encounter vector spaces and basis.
To keep it simple, if I have the color spectrum and the three primary colors — red, blue, and yellow, then the spectrum is the vector space while the primary colors are the basis. Here, we also have to understand that all these primary colors are completely distinct and independent, which in mathematical terms is known as ‘linearly independent’.
A linear combination is just the addition of basis vectors with a scalar. So for the color green, we get a linear combination
To get a clearer understanding of these concepts you can refer 3Blue1Brown videos on the fundamentals of linear algebra.
Wave Function and Its Role in Superposition
In the context of quantum mechanics, we use the wave function to represent quantum particles. The wave function is a linear combination of basis states, much like mixing primary colors to create different shades. Each basis state corresponds to a specific quantum state, and the coefficients in the linear combination represent the probability amplitudes associated with each state.
Understanding Quantum Superposition
Superposition in quantum mechanics is not a simultaneous existence of a particle in multiple positions. Instead, it denotes that the quantum particle exists in a combination of possible states, each with an associated probability amplitude. Until measured or observed, the particle’s exact state remains uncertain.
Mathematically, for a quantum system described by a wave function |ψ⟩, it can be in a superposition of basis vectors, which is called basics states in quantum physics, represented as:
But doesn’t it look like simultaneously being in multiple states? Let’s take an example where I will consider position coordinates, for simplicity.
Imagine a particle described by a wave function |ψ⟩ = α|0⟩ + β|1⟩, where |0⟩ and |1⟩ represent two possible states. It may initially seem as if the particle exists simultaneously in both states, much like a body at the position 5x + 3y might appear to be at both 5x and 3y coordinates simultaneously. However, this is not the case.
In reality, the position 5x + 3y merely represents a point between the linearly independent x and y axes. Similarly, the wave function |ψ⟩ = α|0⟩ + β|1⟩ signifies that the quantum particle exists in a coherent combination of the states |0⟩ and |1⟩ until a measurement is made.
Just as the body’s position is determined by proportions (5 and 3) along the x and y axes, the quantum particle’s state is determined by the complex probability amplitudes α and β associated with each basis state. These amplitudes represent the probabilities of finding the particle in states |0⟩ and |1⟩ upon measurement.
The crucial distinction between classical and quantum systems lies in the measurement process. In classical cases, the function representing a body’s position remains unchanged. However, in quantum mechanics, upon measurement, the wave function collapses to one of the basis states, with probabilities given by the squared magnitudes of the probability amplitudes. This collapse is a fundamental property of quantum mechanics and leads to specific outcomes when measurements are performed.
In summary, the analogy with position coordinates clarifies that superposition does not imply simultaneous existence in multiple states. Instead, it denotes a combination of possible states described by the wave function, with a definitive state emerging only upon measurement. coordinates for simplicity.
Phew!!! That was a lot of math.
Conclusion
Quantum superposition, a truly intriguing phenomenon, serves as the bedrock for the swiftly evolving Quantum Industry. It forms the backbone of Quantum Computing, which harnesses this enigmatic property to achieve remarkable advancements in various fields.
If you want to know a bit about the weird behavior exhibited by particles due to superposition, check out the video by Up and Atom
And if you have more time, you can watch the 1-hour long version of the above video