Number operator

In quantum mechanics, the number operator is an important mathematical operator associated with quantized systems, particularly those with discrete energy levels. The number operator is used to find the expected or average number of particles in a particular quantum state or energy level.

It will be shown that the eigen values of a^+a are 0,1,2,3......So a^+a is known as number operator (N)

N = a^+a

Important relation to be used to find the eigen values of N.

i. Na = a(N-1)

ii. Na+ = a+(N+1)

iii. [ N, a] = - a

iv. [ N, a+] = a+

Eigen values of Number operator

Let l un > be the eigen ket of N with eigen value n

N l un > = n l un >

Now we consider Na l un > = a (N - 1) l un >

= a N l un > - a l un >

= an l un > - al un >

= ( n- 1) a l un >

This indicate that a l un > is also eigen ket of N with eigen value (n-1). Now we calculate square of the norm of al un >

( a l un >, al un > ) = < un l a+a l un > = < un l N l un > = n < un l un > = n

Since square of the norm of a vector cannot be negative

n ⩾ 0

Thus eigen value of number operator can not be negative

We know,

N l un > = n l un >

Na l un > = (n-1) al un >

Also it can be shown that

Na² l un > = (n-2)a² l un >

Na³ l un > = (n-3) a³ l un >

Thus it appears that we can construct norms sequence of eigen kets of N with eigen values n, (n-1), (n-2),......This indicated that eigen values of N can be negative also. But this would conflict with the fact that an eigen value of N can not ne negative. This contradiction can be avoided only if the sequence terminates at zero eigen values of N such that Nl uo > = 0l uo > and al un > = 0.

The state l uo > is known as vacuum state (ground state)

Further application of a will produce only zero vector

a² l uo > = 0

a³ l uo > = 0

There is no chance of negative value of eigen values of N. Thus eigen values of N are 0,1,2,4.....n

Eigen values and eigen kets of H

N l un > = n l un >

We consider Hl un > = ℏ w (N + 1/2 ) l un > = ℏ w N l un > + ℏ w / 2 l un >

= ℏ w n l un > + ℏ w / 2 l un >

= ℏ w ( n + 1/2 ) l un >

Thus un is also eigen ket of H with eigen value ℏ w ( n + 1/2 )

Hl un > = En l un >

En = ℏ w ( n + 1/2)

n= 0, Eo = ℏ w /2 energy in the vaccum state

n = 1, E_1 = ℏ w (3/2) = 3/2 ℏ w = ℏ w + 1/2 ℏ w

n=2, E_2 = ℏ w (5/2) = 5/2 ℏ w = 2 ℏ w + 1/2 ℏ w

n= 3, E_3 = ℏ w (3/2) = 3/2 ℏ w = 3 ℏ w + 1/2 ℏ w

En = nℏ w + 1/2 ℏ w

Thus energy of harmonic oscillator is quantized. ℏw is known as one packet of energy (one quantum of energy)

The number operator is commonly used in quantum systems with a quantized energy spectrum, such as harmonic oscillators, photons in a quantized electromagnetic field, or particles in bound states in atomic or molecular systems. In these cases, energy levels are quantized, and particles can occupy different energy eigenstates with definite energy values.

The number operator plays a crucial role in quantum mechanics, particularly in understanding the statistics of particles in quantized systems. It is closely related to concepts like quantization of energy levels, quantum statistics (such as Bose-Einstein and Fermi-Dirac statistics), and the quantized nature of certain physical observables in quantum theory.

Here's how it works: If you apply the number operator to a specific energy level, it gives you a number. This number tells you how many particles are in that energy level at a given time.

So, just like you'd use a tool to count marbles in a slot, scientists use the number operator to count particles in different energy levels. It's an important idea in quantum mechanics because it helps us understand how particles are distributed among different states and how they behave in quantized systems.

This note is a part of the Physics Repository.