Derivations

# 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^**. Again from Orthogonality theorem, we know that^{j}**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**

**aᵢ**

**²**. 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**.
So, don’t forget to

**subscribe and follow our website (buttons are available in the right sidebar)**and**f****ollow****me on social media**for updates….**To know basics of quantum world, astronomy and space exploration you can check out my book "Through the wormhole" on amazon kindle.**

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-Ratnadeep Das Choudhury

Founder and writer of The Dynamic Frequency

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