In L. Susskind's book "Quantum Mechanics The Theoretical Minimum, spin is used as a vehicle to explain the effect of an observable in three orthogonal diagrams, leading to the creation of the Pauli Spin matrices. My question is the following specific aspect of the reasoning leading to the Pauli Spin matrices.

The notation used is: Along the $z$-axis, we measure the spin states $\lvert u\rangle$ 'up' and $\lvert d\rangle$ 'down'. Along the $x$-axis, we measure $\lvert l\rangle$ 'left' and $\lvert r\rangle$ 'right', and along the $y$-axis, we measure $\lvert i\rangle$ 'in' and $\lvert o\rangle$ 'out'.

We have

$$|r\rangle = \frac{1}{\sqrt{2}}|u\rangle+\frac{1}{\sqrt{2}}|d\rangle\tag{1}$$

$$ \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1\\ \end{pmatrix} $$

$$\sigma_{z}|r\rangle = \frac{1}{\sqrt{2}}|u\rangle-\frac{1}{\sqrt{2}}|d\rangle\tag{2}$$

Equation (1) is derived, see book page 41, equation 2.5, Section 2.3 by using the following setup:

  1. Apparatus A prepares the spin state to be $|r\rangle$

  2. Apparatus A is then rotated to measure $\sigma_{z}$

However this "looks" to be doing the same as the left hand side of equation (2), but the right hand side of equation (2) [derived on page 82] has a minus sign that isn't there in equation (1). So why is there a difference?


2 Answers 2


In page 41, we are only told that measuring $|r\rangle$ gives us $|u\rangle$ and $|d\rangle$ with equal probabilities. Here we are using the measurement of $\sigma_z$ of an electron in $|r\rangle$ state to determine it in terms of $|u\rangle$ and $|d\rangle$. We are only looking at probabilities here. Not probability amplitudes. Thus when forming the state we have the freedom to call either $$|r\rangle=\frac{1}{\sqrt{2}}|u\rangle \pm \frac{1}{\sqrt{2}} |d\rangle$$ as the state. We chose to call the plus state as $|r\rangle$.

However in page 82, we are considering the action of $\sigma_z$ on the state $|r\rangle$ at the level of amplitudes. And since we have already decided on what the state $|r\rangle$ is, we can do this calculation which turns out to be $\frac{1}{\sqrt{2}}|u\rangle - \frac{1}{\sqrt{2}} |d\rangle$.

  • $\begingroup$ Apologies for being slow - why isn't σz = z ? On p41 Susskind creates a state vector |r> that is intended to match the probability that would occur on measurement, which sounds to be the same as what σz is doing on p82 ? Would it be possible to further explain why p41 isn't the same as p82 ? Thanks. $\endgroup$ Jan 19, 2020 at 12:44
  • $\begingroup$ In 41 we talk about the probabilities. But in 82 we talk about amplitudes. $\endgroup$ Jan 19, 2020 at 13:49
  • $\begingroup$ Page 41 does appear to be talking about amplitudes ? It considers the amplitudes needed to set <r,r>=1 and the amplitudes are the numbers in front of |u> and |d> in equation (1). If this isn't what Susskind is doing on p41, then what is p41 doing physically if its not creating a state vector |r> which is the state being prepared (|r>) and then the apparatus is rotated to the z axis, produces the correct probability amplitudes ? $\endgroup$ Jan 19, 2020 at 17:37
  • $\begingroup$ In 41 we are using the fact that we get a 50-50 chance of up or down when measuring right to express it in terms of up and down. Here we can only say that the coefficients must have a magnitude of $\frac{1}{\sqrt{2}}$ and nothing about the relative phase. We have the freedom to choose this phase which for right we take as 0. $\endgroup$ Jan 19, 2020 at 19:07
  • $\begingroup$ In other words, in 41 we are deriving the state by working back at it based on our experimental observation. And in 82 we are using this state to predict the action of other measurable. $\endgroup$ Jan 19, 2020 at 19:10

After the apparatus measured the spin along the +x axis, the particle has the state $|r\gt$.

Applying the operator $ \sigma_z$ doesn't modify that state. It is not a measurement (yet). So:

$$\sigma_{z}|r\gt = \frac{1}{\sqrt{2}}|u\gt-\frac{1}{\sqrt{2}}|d\gt$$

is just the acting of

$$ \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1\\ \end{pmatrix} $$

over the state $|r\gt$. It is a multiplication of that vector by that matrix. The minus sign come from the matrix.

The meaning of that operation is to be able to calculate the probability and expected value of an eventual measurement along +z orientation.

  • $\begingroup$ OK, I see how the operator does its stuff and where the - sign comes from. Do you know why the equation (2) isn't the same as equation (1), which it 'looked' that it should be ? So what have I missed in my reasoning ? $\endgroup$ Jan 17, 2020 at 14:30
  • $\begingroup$ (1) is the state vector. (2) is the result of the operator on it. $\endgroup$ Jan 17, 2020 at 14:33
  • $\begingroup$ OK I agree. However the way of constructing the state vector equation (1) as described in Step (1) and step (2) looks to be the same as applying the operator to it, so if that's true equations (1) and (2) should have the same right hand sides - but they don't. $\endgroup$ Jan 17, 2020 at 15:04
  • $\begingroup$ I don't have the book. But it is the outcome of step (2) (expected value and probability) that needs the operator. The state vector for x+ is correct and it is done after step (1). $\endgroup$ Jan 17, 2020 at 16:15
  • $\begingroup$ The Pauli Spin Operators do the following on spins prepared in the right & left states : σz |r> becomes |left>, whereas σz |left> becomes |right>. The problem is that Susskind 'apparently' uses the same process to create the state vectors in terms of |u> and |d>. Its a shame you dont have the book for without it I dont suppose the question means much. In the next comment I will copy the words deriving equation (1) (Apologies for the formatting) $\endgroup$ Jan 17, 2020 at 17:12

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