In quantum mechanics, "a+" and "a" are operators that are used to describe the behaviour of a quantum system, like the magical bouncing ball we talked about earlier. They're like special tools that help us understand how the system behaves in terms of position and momentum.

Imagine you have a magical box that can create waves that make the ball bounce. The "a+" operator is like the button you press to create a wave that lifts the ball up. And the "a" operator is like the button you press to create a wave that pushes the ball down. These operators are related to how the ball's position and momentum change when those magical waves happen.

These operators help us understand the energy levels and movement of the tiny ball in the quantum world. They're used a lot when studying the harmonic oscillator, which is like a dance that tiny particles do—bouncing back and forth between two points.

So, in a nutshell, "a+" and "a" are special tools that scientists use to understand how things move and change in the quantum world, kind of like how we use buttons to control toys in a game.

N l u_n > = n l u_n>

We consider the state

Na^+ l u_n > = a+ (N+1) l u_n> = a^+N l u_n> + a^+ l u_n> = a^+ n l u_n> + a^+ l u_n> = (n+1) a^+lu_n>

a+ l un> is also eigen ket of N with eigen value (n+1). This indicates that a^+ changes the state having n quanta of energy into the state having (n+1) quanta of energy a^+ creates one quantum of energy. So a+ is known as creation operator.

Na l u_n> = (n-1) a lu_n>

The operator a changes the state having n quanta of energy into the state having (n-1) quanta of energy. Thus the operator a annihilates one quantum of energy. a is known as annihilation operator.

For the given normalized ket l un> is the ket alun > normalized

( a l u_n> , a lu_n>) = < u_n l a^+a l u_n > = <un l N l un> = n <un l un > = n

So in general al un> is not normalized. So the normalized ket of N with eigen value n-1 is given by

So the normalized set N with eigen value (n+1) is given by

So, in summary, "a+" and "a" are like special tools in the quantum toolbox. They let us create and remove particles, which is really important for understanding how things work on the tiniest scales, like the dance of particles in a harmonic oscillator.

This note is a part of the Physics Repository.

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