Timeline for The Theoretical Minimum: Confusion Over Susskind's Reasoning for mutually orthogonal states
Current License: CC BY-SA 3.0
11 events
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| Apr 3, 2016 at 12:48 | comment | added | MonaLisaOverdrive | Still working at this…went back and re-read the section starting with the statement that I added to the question. I did find where Susskind stated that the up/down directions are NOT orthogonal even though associated state vectors are within state space. When speaking of the representations of $|l\rangle$ and $|r\rangle$ in terms of $|u\rangle$ and $|d\rangle$, is it still related to generic state $|A\rangle$? | |
| Mar 22, 2016 at 2:53 | comment | added | MonaLisaOverdrive | The edits to your answer help in that they make it clear that I need to go back and re-read some book sections more carefully; Susskind does not spend a lot of time repeating himself so I need to make sure I know the vector spaces he's referring to when he refers to them | |
| Mar 21, 2016 at 17:45 | comment | added | knzhou | @MonaLisaOverdrive This is why, e.g., you can get "right" by adding together "up" and "down". That makes no sense if you try adding together the spin directions. | |
| Mar 21, 2016 at 17:44 | comment | added | knzhou | @MonaLisaOverdrive When he names the states $u$, $d$, etc., he is using the latter, because the names correspond to the spin directions. But when he relates the states together (like on the right hand side of your first equation), he is using the former, because that deals with the state space structure. | |
| Mar 21, 2016 at 17:42 | comment | added | MonaLisaOverdrive | @knzhou: Which of the two is Susskind referring to in the original post? My hunch is the latter of your two examples (where $|u\rangle$ and $|d\rangle$ are 180 degrees apart) but I'd like a definite answer. | |
| Mar 21, 2016 at 17:37 | comment | added | knzhou | @MonaLisaOverdrive Be careful! There's a distinction between the abstract direction of the state vectors in the vector space (where u and d are 90 degrees apart) and the direction of the spins corresponding to the state vectors in physical space (where the spins corresponding to u and d are 180 degrees apart). You're mixing these two up. To be fair though, this is one of the most confusing things about spin 1/2. | |
| Mar 21, 2016 at 15:39 | history | edited | Javier | CC BY-SA 3.0 |
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| Mar 21, 2016 at 15:25 | comment | added | MonaLisaOverdrive | Well, that's more confusing as the book states that $|u\rangle$ and $|d\rangle$ are basis vectors along $z$... | |
| Mar 21, 2016 at 13:24 | comment | added | Javier | @MonaLisaOverdrive: Remember that kets are abstract vectors. If we take $\{|u\rangle, |d\rangle\}$ as a basis, that means that $|u\rangle$ is the x-axis and that $|d\rangle$ is the y-axis. Draw $|r\rangle$ in a plane and figure out a vector orthogonal to it. | |
| Mar 21, 2016 at 0:13 | comment | added | MonaLisaOverdrive | Sorry, I'm not understanding this either. When I visualize it on a Cartesian plane it kind of makes sense; I see $|l\rangle$ as the negative numbers on the x-axis, and assume that means $|l\rangle = \frac{1}{\sqrt{2}}|u\rangle - \frac{1}{\sqrt{2}}|d\rangle$ but I can't grok it mathematically. I'll revisit it in the morning and hope that helps | |
| Mar 20, 2016 at 18:36 | history | answered | Javier | CC BY-SA 3.0 |