Quantum Harmonic oscillator

  • Thread starter TimeRip496
  • Start date
  • #1
TimeRip496
254
5
upload_2016-1-31_3-3-25.png
upload_2016-1-31_3-33-26.png
upload_2016-1-31_3-3-43.png
upload_2016-1-31_3-4-23.png

When I work out $$b^+b$$, I get

$$\widehat{b^+} \widehat{b} = \frac{1}{2} (ξ - \frac{d}{dξ})(ξ + \frac{d}{dξ}) = \frac{1}{2} (ξ^2 - \frac{d^2}{dξ^2}) = \frac{mωπx^2}{h} - \frac{h}{4mωπ} \frac{d^2}{dx^2}$$

So base on what I have about, (9) should be
$$(9) = \frac{hω}{2π} (\frac{1}{2} \widehat{b^+} \widehat{b} + \frac{1}{2} ξ^2)$$
instead of (10).

Am I missing out something?
 

Attachments

  • upload_2016-1-31_3-2-57.png
    upload_2016-1-31_3-2-57.png
    18.3 KB · Views: 378
Last edited:

Answers and Replies

  • #2
blue_leaf77
Science Advisor
Homework Helper
2,637
785
##\frac{1}{2}(ξ-\frac{d}{dξ})(ξ+\frac{d}{dξ})=\frac{1}{2}(ξ^2-\frac{d^2}{dξ^2})##
The above step is not correct. Instead, it should be
$$
\frac{1}{2}(\xi-\frac{d}{d\xi})(\xi+\frac{d}{d\xi}) = \frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2}).
$$
Then simplify the second and third terms.
 
  • #3
TimeRip496
254
5
##\frac{1}{2}(ξ-\frac{d}{dξ})(ξ+\frac{d}{dξ})=\frac{1}{2}(ξ^2-\frac{d^2}{dξ^2})##
The above step is not correct. Instead, it should be
$$
\frac{1}{2}(\xi-\frac{d}{d\xi})(\xi+\frac{d}{d\xi}) = \frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2}).
$$
Then simplify the second and third terms.
Based on yours,
$$
\frac{1}{2}(\xi-\frac{d}{d\xi})(\xi+\frac{d}{d\xi}) = \frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2}) = \frac{mωπx^2}{h} - \frac{h}{4mωπ} \frac{d^2}{dx^2} -\frac{1}{2} + x\frac{1}{2} \frac{d}{dx}
$$
This is where i am stuck again as to what do I do with $$x\frac{1}{2} \frac{d}{dx}$$ and this one( $$\frac{h}{4mωπ} \frac{d^2}{dx^2}$$) has the term 4 instead of 8 as shown in equation (9).
 
  • #4
blue_leaf77
Science Advisor
Homework Helper
2,637
785
$$
\frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2}) = \frac{mωπx^2}{h} - \frac{h}{4mωπ} \frac{d^2}{dx^2} -\frac{1}{2} + x\frac{1}{2} \frac{d}{dx}
$$
That's still incorrect.
The term ##\frac{d}{d\xi}\xi## should expand into two terms, what are they?
 
  • #5
TimeRip496
254
5
That's still incorrect.
The term ##\frac{d}{d\xi}\xi## should expand into two terms, what are they?
##\frac{d}{d\xi}\xi = \frac{d}{d\xi}(\xi * 1) = 1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}##
Is this correct?
 
  • #6
blue_leaf77
Science Advisor
Homework Helper
2,637
785
##\frac{d}{d\xi}\xi = \frac{d}{d\xi}(\xi * 1) = 1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}##
Is this correct?
Yes, that one is correct.
 
  • #7
TimeRip496
254
5
Yes, that one is correct.
What about this one( $$\frac{h}{4mωπ} \frac{d^2}{dx^2}$$) which is supposed to be in equation (9)?

The only way I can get to $$\frac{h^2}{8mωπ^2} \frac{d^2}{dx^2}$$ is by multiplying $$\frac{h}{4mωπ} \frac{d^2}{dx^2}$$ with $$\frac{h}{2π}$$, which I don't know where to get from.






How I get $$\frac{h}{4mωπ} \frac{d^2}{dx^2}$$,
First, $$dξ = \sqrt[] {\frac{2mπω}{h}} * dx$$
Squaring it, $$dξ^2 = \frac{2mπω}{h} * dx^2$$
Subbing it into $$ \frac{d^2}{dξ^2} $$ from b+*b, I get $$\frac{h}{2mωπ} \frac{d^2}{dx^2}$$ before multiplying 0.5 from the bracket outside $$\frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2})=b^+*b$$.
 
Last edited:
  • #8
blue_leaf77
Science Advisor
Homework Helper
2,637
785
upload_2016-1-31_3-3-43-png.95071.png
The coefficient in front of the second order derivative in the above equation is not correct, as you can check its dimensionality. It should be
$$
\frac{h}{4m\omega \pi}
$$
which is the result you obtained from your own calculation. This is the correct one as you can check by multiplying it with the constant outside the square bracket and it should lead to ##p^2/(2m)##.
 
  • #9
TimeRip496
254
5
##\frac{1}{2}(ξ-\frac{d}{dξ})(ξ+\frac{d}{dξ})=\frac{1}{2}(ξ^2-\frac{d^2}{dξ^2})##
The above step is not correct. Instead, it should be
$$
\frac{1}{2}(\xi-\frac{d}{d\xi})(\xi+\frac{d}{d\xi}) = \frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2}).
$$
Then simplify the second and third terms.
The coefficient in front of the second order derivative in the above equation is not correct, as you can check its dimensionality. It should be
$$
\frac{h}{4m\omega \pi}
$$
which is the result you obtained from your own calculation. This is the correct one as you can check by multiplying it with the constant outside the square bracket and it should lead to ##p^2/(2m)##.
Thanks a lot!
 
  • #10
TimeRip496
254
5
##\frac{1}{2}(ξ-\frac{d}{dξ})(ξ+\frac{d}{dξ})=\frac{1}{2}(ξ^2-\frac{d^2}{dξ^2})##
The above step is not correct. Instead, it should be
$$
\frac{1}{2}(\xi-\frac{d}{d\xi})(\xi+\frac{d}{d\xi}) = \frac{1}{2}(\xi^2 +\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi - \frac{d^2}{d\xi^2}).
$$
Then simplify the second and third terms.
Sorry to trouble you again.
Talking about the second and third terms, $$ \xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi = \xi\frac{d}{d\xi} - (1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}) $$
Initially I thought I can cancel out the $$\xi\frac{d}{d\xi}$$ by subtracting it with $$\xi * \frac{d1}{d\xi}$$, but in my another thread, you told me that
$$∅*(\frac{d}{dξ})\neq \frac{d∅}{dξ}$$, thats why I am stuck at this part.
 
  • #11
blue_leaf77
Science Advisor
Homework Helper
2,637
785
I should probably have corrected your post #5, instead of equal sign it might have been more appropriate to use arrow instead:
$$
\frac{d}{d\xi}\xi \rightarrow \frac{d}{d\xi}(\xi * 1) = 1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}.
$$
with the arrow sign is intended to mean that the expression to the left is to be operated on a function, in the above case a number ##1##. The insertion of a ##1## there is supposed to be thought as an arbitrary object because remember that Hamiltonian is actually an operator, it's not yet a physical quantity. Therefore, writing ##H## only with nothing following it does not have mathematical significance. Only when you make it operate on an arbitrary state ##\psi## will the expression ##H\psi## be meaningful.
##
\xi\frac{d}{d\xi} - \frac{d}{d\xi}\xi = \xi\frac{d}{d\xi} - (1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi})##
In the above equation, if you make it as if it operates on a ##1## (or an arbitrary state ##\psi##), it should become
$$
\xi\frac{d}{d\xi}1 - (1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}).
$$
 
  • #12
TimeRip496
254
5
I should probably have corrected your post #5, instead of equal sign it might have been more appropriate to use arrow instead:
$$
\frac{d}{d\xi}\xi \rightarrow \frac{d}{d\xi}(\xi * 1) = 1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}.
$$
with the arrow sign is intended to mean that the expression to the left is to be operated on a function, in the above case a number ##1##. The insertion of a ##1## there is supposed to be thought as an arbitrary object because remember that Hamiltonian is actually an operator, it's not yet a physical quantity. Therefore, writing ##H## only with nothing following it does not have mathematical significance. Only when you make it operate on an arbitrary state ##\psi## will the expression ##H\psi## be meaningful.

In the above equation, if you make it as if it operates on a ##1## (or an arbitrary state ##\psi##), it should become
$$
\xi\frac{d}{d\xi}1 - (1 * \frac{d\xi}{d\xi} + \xi * \frac{d1}{d\xi}).
$$
Thanks a lot again! One last thing if you wouldn't mind,
After converting equation (9) to equation (10),
$$\hat{b}*∅_0 = 0$$
Thus, $$\hat{b}^+\hat{b}*∅_0 = 0$$,
So equation (10) left with $$\frac{hω}{2π}(0+\frac{1}{2})∅_0=E_0∅_0$$
Shouldn't $$E_0=\frac{hω}{4π}$$ instead of $$E_0=\frac{hω}{2π}$$?
 
  • #13
blue_leaf77
Science Advisor
Homework Helper
2,637
785
Doesn't the equation ##\frac{hω}{2π}(0+\frac{1}{2})∅_0=E_0∅_0## correctly imply that ##E_0=\frac{hω}{4π}##?
 
  • #16
TimeRip496
254
5
I think you should look for another source to study this subject, the author seems to be rather sloppy with his derivation.
Ok thanks!
 

Suggested for: Quantum Harmonic oscillator

Replies
1
Views
497
Replies
5
Views
603
Replies
21
Views
985
Replies
2
Views
488
  • Last Post
Replies
4
Views
505
  • Last Post
Replies
1
Views
422
Replies
1
Views
670
Replies
4
Views
583
Top