Commutation relation of the position and momentum operators

  • #1

Homework Statement



I've just initiated a self-study on quantum mechanics and am in need of a little help.

The position and momentum operators do not commute. According to my book which attemps to demonstrate this property,
(1) [tex]\hat{p} \hat{x} \psi = \hat{p} x \psi = -i \hbar \frac{\partial}{\partial x}(x \psi)[/tex]

OK, so far I was following. I'm not understaing how they get to this next equation from the previous one...
(2) [tex] = -i \hbar \left( \psi + x \frac{\partial \psi}{\partial x}\right)[/tex]

To be more precise, where did the second psi symbol come from and where did the + symbol come from in equation 2?

Homework Equations



[tex] \hat{x} = x [/tex]
[tex] \hat{p} = -i \hbar \frac{\partial}{\partial x} [/tex]


The Attempt at a Solution



When I attempt to multiply the momentum and position operators, I get:
(3) [tex] \hat{p} \hat{x} \psi = \hat{p} x \psi = \left( -i \hbar \frac{\partial}{\partial x} \right) x \psi [/tex]

But how do I go from equation 3 and arrive at equation 2? Or can I?
 
Last edited:

Answers and Replies

  • #2
Meir Achuz
Science Advisor
Homework Helper
Gold Member
3,529
110
psi also depends on x. This leads to the second derivative.
You may need a live teacher to learn QM.
 
  • #3
psi also depends on x. This leads to the second derivative.
You may need a live teacher to learn QM.
OK, I understand the first part, but I am not following why it "leads to the second derivative." Could you please explain in more detail? Or preferably, could you show me the next few steps after equation (1)?
 
  • #4
dextercioby
Science Advisor
Homework Helper
Insights Author
13,005
554
What book are you using ? Throw it away. One postulates the commutation relations, they cannot be proved.
 
  • #5
What book are you using ? Throw it away. One postulates the commutation relations, they cannot be proved.
The author of the book is demonstrating that [tex] \hat{p} \hat{x} \psi does NOT equal \hat{x} \hat{p} \psi [/tex] .

Sorry if I miscommunicated this point in the first post.
 
  • #6
I understand the solution to why [tex] \hat{x} \hat{p} \psi [/tex] equals what it equals. My question just has to do with how the author gets the solution that he does for the [tex] \hat{p} \hat{x} \psi [/tex]
 
  • #7
cristo
Staff Emeritus
Science Advisor
8,107
73
Are you asking how you get from the left to the right hand side of this? [tex]\left( -i \hbar \frac{\partial}{\partial x} \right) x \psi = -i \hbar \left( \psi + x \frac{\partial \psi}{\partial x}\right)[/tex].

If so, then note that [itex]x\psi[/itex] is a product, each of which is dependent on x, and so we must use the product rule. So, starting from the left [tex]\left( -i \hbar \frac{\partial}{\partial x} \right) x \psi =-i\hbar\frac{\partial}{\partial x}(x\psi)=-i\hbar(x\frac{\partial\psi}{\partial x}+\psi)[/tex]
 
Last edited:
  • #8
nrqed
Science Advisor
Homework Helper
Gold Member
3,736
279

Homework Statement



I've just initiated a self-study on quantum mechanics and am in need of a little help.

The position and momentum operators do not commute. According to my book which attemps to demonstrate this property,
(1) [tex]\hat{p} \hat{x} \psi = \hat{p} x \psi = -i \hbar \frac{\partial}{\partial x}(x \psi)[/tex]

OK, so far I was following. I'm not understaing how they get to this next equation from the previous one...
(2) [tex] = -i \hbar \left( \psi + x \frac{\partial \psi}{\partial x}\right)[/tex]

To be more precise, where did the second psi symbol come from and where did the + symbol come from in equation 2?
It's simply the product rule. If [itex] f (x) [/itex] is a function of x, then [tex] \frac{d}{dx} ( x f(x) ) = 1 \times f(x) + x \times \frac{df(x)}{dx} [/tex]
, right?

What book are you using? I would highly recommend Shankar and, as a supplement, Griffiths.
 
  • #9
Great. Thanks a LOT cristo and nrqed!

I love this site! :)

What book are you using? I would highly recommend Shankar and, as a supplement, Griffiths.
I am currently using two books: "Quantum Mechanics for Undergraduates" by Norbury as well as Griffiths (both books use the same template and complement each other strongly). I also have Shankar, however, after going through the first chapter, I found it was too math heavy... perhaps it is aimed at graduates?
 

Related Threads on Commutation relation of the position and momentum operators

Replies
4
Views
2K
Replies
4
Views
5K
Replies
2
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
7
Views
3K
Top