Why Are My Tensor Products Not Adding Up Correctly?

[Jim Kata]

Say I wanted to tensor |1,-1> \otimes |1,0> Then looking at the Clebsch Gordons I get |1,-1> \otimes |1,0> = \frac{1}{\sqrt {2}}|2,-1> - \frac{1}{\sqrt{2}}|1,-1>
When I try to do this another way I run into a problem that I don't understand.

|1,0> = \frac{1}{\sqrt{2}} (|\frac{1}{2}, \frac{1}{2}> \otimes |\frac{1}{2},-\frac{1}{2}> + |\frac{1}{2},-\frac{1}{2}> \otimes |\frac{1}{2},\frac{1}{2}>)

So

|1,-1> \otimes |1,0> = |1,-1> \otimes ( \frac{1}{\sqrt{2}} (|\frac{1}{2}, \frac{1}{2}> \otimes |\frac{1}{2},-\frac{1}{2}> + |\frac{1}{2},-\frac{1}{2}> \otimes |\frac{1}{2},\frac{1}{2}>))

Going through this I get

|1,-1&gt;\otimes|\frac{1}{2},\frac{1}{2}&gt;\otimes|\frac{1}{2},-\frac{1}{2}&gt;<br /> =(\frac{1}{\sqrt{3}}|\frac{3}{2},-\frac{1}{2}&gt; -\sqrt{\frac{2}{3}}|\frac{1}{2},-\frac{1}{2}&gt;)\otimes|\frac{1}{2},-\frac{1}{2}&gt;=\frac{1}{2}|2,-1&gt; +(\frac{\sqrt{3}}{6}-\sqrt{\frac{2}{3}})|1,-1&gt;

similarly I get

|1,-1&gt;\otimes|\frac{1}{2},-\frac{1}{2}&gt;\otimes|\frac{1}{2},\frac{1}{2}&gt;=\frac{\sqrt{3}}{2}|2,-1&gt;-\frac{1}{2}|1,-1&gt;

but when I add these two I don't get

|1,0&gt; = \frac{1}{\sqrt{2}} (|\frac{1}{2}, \frac{1}{2}&gt; \otimes |\frac{1}{2},-\frac{1}{2}&gt; + |\frac{1}{2},-\frac{1}{2}&gt; \otimes |\frac{1}{2},\frac{1}{2}&gt;)

What am I doing wrong? Where does my reasoning break down?

 

[DrClaude]

similarly I get

|1,-1&gt;\otimes|\frac{1}{2},-\frac{1}{2}&gt;\otimes|\frac{1}{2},\frac{1}{2}&gt;=\frac{\sqrt{3}}{2}|2,-1&gt;-\frac{1}{2}|1,-1&gt;

That's incorrect,
$$
|1,-1 \rangle \otimes|\frac{1}{2},-\frac{1}{2}\rangle\otimes|\frac{1}{2},\frac{1}{2}\rangle= \frac{1}{2} |2,-1\rangle - \frac{\sqrt{3}}{2}|1,-1 \rangle
$$