# Raising momentum operator acting on spherical harmonics

• FourierX
In summary, the momentum operator in quantum mechanics is a representation of momentum and is denoted by the symbol p. It acts on a wave function by multiplying it by the momentum value at a specific point in space. Spherical harmonics are special functions used to describe the wave function of a particle in a spherical potential, and they are related to the momentum operator through the spherical harmonics operator. The significance of the raising momentum operator acting on spherical harmonics lies in its ability to determine the momentum of a particle in a given state and obtain the next higher state of the particle. This is important in understanding the behavior of particles in a spherical potential and solving quantum mechanical problems.
FourierX

## Homework Statement

What is the result of raising momentum ladder operator (L+) acting on spherical harmonics Y04 ($$\theta$$,$$\phi$$)

## The Attempt at a Solution

I was expecting Y14 ($$\theta$$,$$\phi$$)

I applied L+ on Y04 ($$\theta$$,$$\phi$$) and ended up with Y14 ($$\theta$$,$$\phi$$) multiplied by $$\sqrt{20}$$h-bar.

That is what you should get:

$$L_+Y_\ell^m=\hbar\sqrt{(\ell-m)(\ell+m+1)}Y_\ell^{m+1}$$

with $\ell=4$ and $m=1$, you get

$$L_+Y_4^0(\theta,\phi)=\hbar\sqrt{20}Y_4^1(\theta,\phi)$$

## 1. What is the momentum operator in quantum mechanics?

The momentum operator in quantum mechanics is a mathematical representation of the physical quantity of momentum. It is denoted by the symbol p and is defined as the product of the mass and velocity of a particle. In quantum mechanics, the momentum operator is a Hermitian operator, meaning its eigenvalues are real numbers and its eigenvectors are orthogonal.

## 2. How does the momentum operator act on a wave function?

The momentum operator acts on a wave function by multiplying it by the momentum value at a specific point in space. In other words, it is applied to the position coordinates of the wave function to determine the momentum of the particle at that location. This is represented by the mathematical expression p= -i(h/2π)∇, where h is Planck's constant and ∇ is the nabla operator.

## 3. What are spherical harmonics in quantum mechanics?

Spherical harmonics are a set of special functions that are used to describe the wave function of a particle in a spherical potential. They are solutions to the Laplace equation in spherical coordinates and have specific angular momentum values. In quantum mechanics, they are used to describe the orbital motion of electrons in an atom.

## 4. How is the momentum operator related to spherical harmonics?

The momentum operator is related to spherical harmonics through the spherical harmonics operator, which is a combination of the position and momentum operators. It is used to describe the angular momentum of a particle in a spherical potential and is represented by the mathematical expression L= -i(h/2π)r x ∇, where r is the position vector.

## 5. What is the significance of raising momentum operator acting on spherical harmonics?

The raising momentum operator acting on spherical harmonics is significant because it allows us to determine the momentum of a particle in a given state. By applying the raising operator to a spherical harmonic, we can obtain the next higher state of the particle, which has a higher momentum value. This is important in understanding the behavior of particles in a spherical potential and in solving quantum mechanical problems.

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