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We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

E D I T: How to do it with more basis vectors? For example in a situation like this (this should have 6 parameters, but I don't know what to do, since I can't use the sine-cosine trick now)

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

E D I T: How to do it with more basis vectors? For example in a situation like this

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

E D I T: How to do it with more basis vectors? For example in a situation like this (this should have 6 parameters, but I don't know what to do, since I can't use the sine-cosine trick now)

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

4 deleted 196 characters in body
source | link

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

E D I T: How to do it with more basis vectors? For example in a situation like this

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

E D I T: How to do it with more basis vectors? For example in a situation like this

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

3 deleted 196 characters in body
source | link

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

EDIT: How to do similar tricks with more basis vectors? For example in a situation like this

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

EDIT: How to do similar tricks with more basis vectors? For example in a situation like this

\begin{align} \ |{\Psi}\rangle=a_a|A\rangle+a_b|B\rangle+a_c|C\rangle+a_d|D\rangle \end{align}

We have a spin state

\begin{align} \ |{\Psi}\rangle=a_u|{U}\rangle+a_d|{D}\rangle \end{align}

where $|U\rangle$ and $|D\rangle$ are $up$ and $down$ basis vectors, and $a_u$,$a_d$ are their complex coefficients.

\begin{align} \ a_u=x+yi \end{align}

\begin{align} \ a_d=w+zi \end{align}

$x, y, w, z$ are the real parameters I'm asking about. Since $a_ua^*_u + a_da^*_d=1$ we can ignore one of these, let's say $z$ and still calculate the state of the system, so now we have three needed parameters.

Now, according to my book, "ignoring the overall phase factor" makes it possible to specify the spin state using only two parameters. I know what the phase factor is and that we can ignore it in the calculations, but I don't see how it relates to the number of parameters needed.

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