# Quick question: Momentum operator in QM

• Libohove90
In summary, the conversation discusses two ways to write the momentum operator, p = (-i hbar d/dx) and p = (hbar / i)d/dx, and how to go from one to the other. The individual tries to solve the equation by squaring both sides, but realizes that -i is not equivalent to i squared. The solution is explained using complex numbers and the exponential form of i.
Libohove90

## Homework Statement

There are two ways to write the momentum operator, p = (-i hbar d/dx) and p = (hbar / i)d/dx. How do you go from one to the other?

## Homework Equations

The two I gave above.

## The Attempt at a Solution

I tried to see if -ih = h/i by squaring both sides, but one came out positive and the other negative. Thanks for the help!

-i^2 = 1, not -1. This makes (-i hbar)^2 not equivalent to (hbar / i)^2

Last edited:
Since when did $(-i)^2 = 1$?

Libohove90 said:
-i^2 = 1, not -1. This makes (-i hbar)^2 not equivalent to (hbar / i)^2

$(-i)^2 = (-1)^2(i)^2$

$i^{-1} =- i$
Why?
$$i^{-1} = e^{-ln(i)}=e^{-ln(e^{i\pi /2})}=e^{-i\pi /2}=-i$$
since $e^{\pm i\pi /2}= cos(\pm\pi /2) + isin(\pm\pi /2)=\pm i$

Or you just multiply 1/i by i/i to get i/i^2 = i/(-1) = -i.

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

The momentum operator in quantum mechanics is a mathematical representation of the physical concept 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 an important mathematical tool used to describe the motion of particles and their interactions.

## 2. How is the momentum operator related to the uncertainty principle?

According to the uncertainty principle, it is impossible to know the exact position and momentum of a particle simultaneously. The momentum operator plays a crucial role in this principle, as it is used to calculate the uncertainty in the momentum of a particle. This uncertainty is inversely proportional to the uncertainty in the position of the particle, as described by the famous Heisenberg Uncertainty Principle.

## 3. What is the mathematical expression for the momentum operator in quantum mechanics?

The mathematical expression for the momentum operator is given by the differential operator p = -iħ∇, where ħ is the reduced Planck's constant and ∇ is the del operator. This expression is used to calculate the momentum of a particle in quantum mechanics.

## 4. How is the momentum operator used in the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum system over time. The momentum operator is used in this equation to calculate the kinetic energy of a particle, which is an essential component of the total energy of the system. This allows us to solve for the wave function of the system and determine its behavior over time.

## 5. Can the momentum operator be measured in experiments?

No, the momentum operator itself cannot be measured in experiments. It is a mathematical representation of the physical concept of momentum and is used in calculations and equations in quantum mechanics. However, the momentum of a particle can be measured experimentally, and the results can be used to validate the predictions made using the momentum operator.

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