- Origin of Quantum Physics
- Wave function
- Collapse of Wave Function
- Physically Accepted Wave function
- Normalization Explained
- Method of Normalization
- Orthogonality & Orthonormality
- Hilbert Space
- Quantization Rules
- Operator Formalism
- Commutator Bracket
- Linear Operator
- Hermitian Operator
- Projection Operator
- Unitary Operator
- Parity Operator
- Expectation Value
- Schrodinger Equation
- Wave-Particle Duality Using Schrodinger Equation
- Superposition of States
- Various Representations of Wave Function
- Probability Current Density
- Uncertainty in Operators
- Shortcut for Calculating Momentum Expectation Value

Einstein summation convention is a mathematical tool for simplifying expressions (bigger to smaller) including summations of vectors, matrices, and general tensors.

Here, we use indices to define a particular term.

i.e.

for example: $a_{i}$ (or $a^{i}$) , where “i” is called index.

As lots of mathematical expressions involve summation over particular indices, that’s why these conventions are used widely to write bigger expressions.

**Rule #1 **

Any twice-repeated index in a “single term” is summed over the positive integers (i=1,2,3…n), usually in n=3

**Explaination:**

If we have the following type of long & boring expression,

$\sum_{i=1}^{3}a_{i}x_{i}= a_{1}x_{1}+ a_{2}x_{2}+ a_{3}x_{3}$

then, using einstein summation convention, we can simply write:

$\sum_{i=1}^{3}a_{i}x_{i} = a_{i}x_{i}$ (where, $a_{i}x_{i}$ is called a “term”)

**Rule #2**

Each index can appear maximum 2 times in a term.

**Explanation:**

Example:

$a_{ij}b_{ij}$ is valid

but, $a_{ii}b_{ij}$ is invalid becuase i is appearing 3 times.