In the lectures by Prof. Leonard Susskind, he mentioned that the total angular momentum squared operator can be represented by  $$ L^2 = L_z^2 + L_z + L_- L_+ $$  ($L_+, L_-$ being ladder operators). He then used it to show the $L^2$ of $|m\rangle$ is $m(m+1)$  ($m$ stands for magnetic quantum number). Finally, he proposed that if $L_-$ commutes with $L^2$ (which is true) then all $(2L+1)$ states will have same $L^2$. However if I apply $L^2 = L_z^2 + L_z + L_- L_+$ to $|m-1\rangle$ directly, it will give me $m^2-m+1 != m(m+1)$  I would like to know where is the flaw in my calculation.
I am watching Susskinds's lecture , in which he introduced square angular momentum operator $L^2$, which can be written as $$ L^2=L_z^2+L_-L_++L_z $$ where $$ L_\pm=L_x\pm iL_y $$  Besides, he argued that the spectrum of $L$ can only be $m,m-1,\cdots,-m+1,-m$ in which $m$ is an integer or half-integer. If we denote $|m\rangle$ the eigenvector corresponds to eigenvalue $m$, we have $$ L_\pm|l\rangle=|l\pm 1\rangle\;\text{or}\;0 $$  In the lecture he showed that $L^2|m\rangle=m(m+1)|m\rangle$ for $m$ the highest eigenvalue, by $$ \begin{align*} L^2|m\rangle&amp;=L_z^2|m\rangle+L_-L_+|m\rangle+L_z|m\rangle\\ &amp;=m^2|m\rangle+0+m|m\rangle\\ &amp;=m(m+1)|m\rangle \end{align*} $$  Then he left an exercise to show that for any $-m\leqslant l&lt; m$, we also have $|l\rangle$ is an eigenvector of $L^2$ with eigenvalue $m(m+1)$.  In the lecture, he hinted by showing that it suffices to show that $L^2$ commutes with $L_i$ for $i=x,y,z$, which gives us $$ L^2|m-1\rangle=L^2L_-|m\rangle=L_-L^2|m\rangle=L_-m(m+1)|m\rangle=m(m+1)L_-|m\rangle=m(m-1)|m-1\rangle $$  I believe it is true since I find the   online.   However, my question is that, why the following proof is wrong.  I tried to mimic the proof for $|m\rangle$ to have that  $$ \begin{align*} L^2|k\rangle&amp;=L_z^2|k\rangle+L_-L_+|k\rangle+L_z|k\rangle\\ &amp;=k^2|k\rangle+L_-L_+|k+1\rangle+k|k\rangle\\ &amp;=k^2|k\rangle+|k\rangle+k|k\rangle\\ &amp;=(k^2+k+1)|k\rangle \end{align*} $$  which shows that $|k\rangle$ does not have eigenvalue $m(m+1)$ but $k^2+k+1$ for $k&lt;m$.
Analysing Bounce Rate and Time on Page VAR/VECM/ARDL optimal lag selection Question 1: Is it necessary to consider AIC and the BIC criteria when selecting the lag for a VAR, VECM or ARDL model OR can I use something else?  Example: Can I pick 12 lags because the model simply produce great results when I do or do I have to use the AIC/SIC criteria?  Question 2: A simple google search tells me that its possible to select lag 4 when quarterly data, 12 when monthly etc. Is that a valid approach ? Was Hogwarts named such because of a connection to warthogs? Was Hogwarts named 'Hogwarts' because of a connection to warthogs?  Or was it named for hogs and warts separately, or something else entirely? In many countries an application for a PhD position includes a written research proposal, so my questions is what are some advises/strategies to come up with a good topic/idea for a PhD research proposal and how can one assess the quality/fruitfulness of an idea? 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