The Quantum Superposition Theorem: A Mathematical Approach

In my earlier articles, I have written mostly about the theoretical side of the story of Quantum Superposition. Quantum superposition is one of the main corner-stone theory which gives the weirdness to Quantum physics and helps us to solve critical problems like Quantum tunneling.

In my last article, I wrote about the Orthogonality theorem which is a necessary prerequisite to understand the mathematics behind Quantum Superposition. Along with this, preliminary calculus with knowledge of probability is required to understand the following text as it may not seem to be that fancy as you see in documentaries rather if you understand the text then it is more fascinating and support the statement: “Facts are stranger than Fiction”.

So, without much further delay, let’s dive into it…..

To formulate superposition principle, firstly we have to consider some potential V(x) and for this potential Schrodinger’s equation has been solved. This yield a number of wave-functions 𝜓ᵢ(x) and their corresponding Eigenvalues for energy Eᵢ.

Now, we have to combine each individual solution by linear summation to construct another function 𝚿(x) and called it as ‘Super-Wave Function’.

𝚿(x) = 𝚺ᵢ aᵢ𝜓ᵢ(x)                   (1)

Here, aᵢ are the expansion coefficients. The expansion coefficients, aᵢ represent the degree to which the full wavefunction possesses the character of the eigenfunction 𝜓ᵢ. Also, by analogy, you can say that the coefficients are the projections of the wavefunction on to the eigenfunctions of the operator.

Normalization of 𝚿(x) over the range of 𝜓ᵢ(x) leads us to,

∫ 𝚿 𝚿*dx = 1

∫ (𝚺ᵢ aᵢ𝜓ᵢ)( 𝚺ⱼ aⱼ*𝜓ⱼ*) = 1      (2)

Now we can swap order if the integration and summation in as it possible without loss of generality in equation 2. So, after swapping we get,

𝚺ᵢ 𝚺ⱼ aᵢ aⱼ* ⟨𝜓ᵢ | 𝜓ⱼ*⟩ = 1      (3)

Here, after rearranging, wave functions for i and j has been represented by Dirac notation. If you know about the Orthogonality theorem, then you must be familiar with this notation. If you don’t know about it, then I strongly recommend you to read it first and then proceed forward.

So, after the above steps, we can use Orthogonality theorem to simplify the equation 3. As we know that, ⟨𝜓ᵢ | 𝜓ⱼ*⟩ can be simplified to ẟᵢ^^j. Again from Orthogonality theorem, we know that ẟᵢ^ ^j is 1 only when i=j. For all other conditions, it is 0. Also, aᵢ aᵢ* = aᵢ².

So, now our equation 3 is modified to,

𝚺ᵢ aᵢ² = 1         (4)

Now, we need to calculate expectation value of 𝚿(x) for our further investigation….

⟨E⟩ = ∫ 𝚿(H 𝚿*)dx = ∫(𝚺ᵢ aᵢ𝜓ᵢ){ H 𝚺ⱼ aⱼ*𝜓ⱼ*} dx      (5)

Here, we used Hamiltonian operator because the general equation of expectation value demands an operator, and in the case of expectation value of energy the operator is Hamiltonian (H).

Also, from basic quantum physics, we know,

        H(𝜓ᵢ) = Eᵢ 𝜓ᵢ                (6)

Now, from eq. 5 and 6,

⟨E⟩ = ∫{ (𝚺ᵢ aᵢ𝜓ᵢ)( 𝚺ⱼ Eⱼ aⱼ*𝜓ⱼ*) } dx     

     

Again we have to follow the same procedure as we did to reduce eq. (2) to eq. (3)

⟨E⟩ = 𝚺ᵢ 𝚺ⱼ Eⱼ aᵢ aᵢ* ⟨𝜓ᵢ | 𝜓ⱼ*⟩

And by Orthogonality theorem,

⟨E⟩ = 𝚺ᵢ Eᵢ aᵢ²          (7)

So, from equation 7, we understand that the expectation value of energy for the superposition state is just the weighted sum of the energies of the individual energies and multiplied with the square of the expansion coefficients (weighted factors).

There are two major postulates of the quantum mechanics and on the basis of these postulates we can constitute the superposition theorem. They are as follows:

1) If a system can be in any of the individual eigenstates 𝜓ᵢ, then it can also be in the superposition state as described by equation 1. Thus, the system need not to be in an eigenstate and may vary.

2) When someone proceeds to make a measurement to determine the state of the system, then the super-wave function or superposition state (𝚿) collapses into one of the eigenstates 𝜓ᵢ. To find the probability of the system in state 𝜓ᵢ can be calculated by using aᵢ². That’s why you can see equation 4 has value 1 as the summation of all probabilities is always 1.

In my further articles, you will be introduced to the formulation of the mathematical structure of the advance theories like perturbation and variation methods. These theories are the basis of the approximation methods which are used in quantum mechanics.

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                                                                 -Ratnadeep Das Choudhury

                                                  Founder and writer of The Dynamic Frequency