Momentum in Classical mechanics and Quantum mechanics

[Dhamnekar Win,odd]

Momentum is used to sense the amount of force applied to a moving object. With the help of Momentum, you can know the nature of the applied force on an object.

Momentum is usually represented by the product of the mass and velocity of a moving object. But in this case of quantum mechanics, the equation of momentum will be different.

 How is this computed?

How is this momentum computed in Quantum mechanics?

[studiot]

Your second equation is incomplete.

i is a constant, hbar is a constant, and nabla (del) is an operator, which needs something to operate on.

You have not provided anything for it to oeprate on.

[Dhamnekar Win,odd]

In the above second question, [math] \nabla r , \hbar [/math]  means gradient r (position vector) and reduced Plank's constant respectively.

Edited November 27, 2021 by Dhamnekar Win,odd

[joigus]

Early quantum theory departed from the idea that energy states of particles in a confined space had to do with wave modes in that confined space (think of a box). Those wave modes are quantized in energy (and therefore in momentum, for free particles) when these waves are in a perfectly reflecting box. It took some time to realise that this could be formulated as due to the fact that matter, in all its forms, has a wave-like nature to it.

The particular hypothesis that answers your question is due to De Broglie. Following De Broglie, a monochromatic wave of wavelength \( \lambda \) has a momentum (think just one possible direction),

\[ p=\frac{h}{\lambda} \]

Think also of free particles. Now, because a harmonic wave of wavelength \( \lambda \) and time period \( T \) is represented by,

\[ \psi\left(x,t\right)=A\cos\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right) \]

and as Studiot said, your “nabla” (derivatives, one for every direction, that in this case is just one, \( x \) ) have to act (differentiate) on this something (the wave),

\[ \frac{h}{2\pi i}\frac{\partial}{\partial x}\psi\left(x,t\right)=2\pi\frac{h}{2\pi i}\frac{1}{\lambda}\sin\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right)=\frac{1}{i}\frac{h}{\lambda}\sin\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right) \]

Now, it just so happens that in quantum mechanics you must complete the so-called wave function so that it has an imaginary part. This imaginary part (for the complex-number version of a monochromatic wave) is,

\[ i\sin\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right) \]

The whole simplest version of this 'wave function' would be,

\[ \psi\left(x,t\right)=A\cos\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right)+i\sin\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right) \]

Repeating the previous process for the whole complex (real plus imaginary parts) leads to,

\[ \frac{h}{2\pi i}\frac{\partial}{\partial x}\psi\left(x,t\right)=2\pi\frac{h}{2\pi i}\frac{1}{\lambda}\sin\left(2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right)=\frac{1}{i}\frac{h}{\lambda}i\psi\left(x,t\right)=\frac{h}{\lambda}\psi\left(x,t\right) \]

So that's what your momentum operator does on states. \( \psi \) is called 'state of a quantum system' or 'wave function'; and the things to measure (momentum) are mathematical operators that extract this information from the state.

I hope that was helpful.

[swansont]

3 hours ago, Dhamnekar Win,odd said:

In the above second question, ∇r,ℏ   means gradient r (position vector) and reduced Plank's constant respectively.

Yes it’s the gradient, and to rephrase what studiot pointed out, the gradient of what? It’s not the gradient of r, it operates on the wave function.

As to your question, what does the wave function of a variable-mass system look like?

[studiot]

7 hours ago, Dhamnekar Win,odd said:

In the above second question, ∇r,ℏ   means gradient r (position vector) and reduced Plank's constant respectively.

 

Since you have made some attempt to answer my comment this time I will offer some more help.

It's clear that you have copy/pasted the expressions in question from a book or other source which should have provided proper background for the questions and the answers already provided.

I don't know what resources you have access to but you are reaching a long way into theoretical physics/applied maths here.
This subject really belongs in one of those two sub forums, not Analysis and Calculus.

Anyway these three attachments should provide a good solid background for both your questions as they compares classical and quantum momenta and operators.