.. _chapter1:
Chapter 1: Stern-Gerlach Experiments
++++++++++++++++++++++++++++++++++++
.. note::
*The hardest part of learning something new is not embracing
new ideas, but letting go of old ones.* ― Todd Rose, The End of
Average: How We Succeed in a World That Values Sameness
Summary
-------
.. attention::
The first steps we take in our exploration of the quantum
mechanical world bring us at the doorstep of a key experiment,
whose outcome cannot be explained by classical physics alone. The
experiment was devised by Stern and Gerlach to study the intrinsic
angular momentum of electrons. Stern-Gerlach (SG) experiments
demonstrate that particles (*i.e.*, electrons) do not behave
classically.
First, the experiments indicate that the particles possess an
angular momentum that is not classical (a classical momentum is
called an *orbital angular momentum* arising from a particle that
rotates, for example, as we will study in
:ref:`chapter10`). Judging from the experimental results, the
*spin* angular momentum seems to present some properties of the
classical angular momentum but also displays a drastically distinct
behavior. For instance, its projection on a given axis takes
*discrete* values (in our example, two of them only, as demonstrated
in the first SG experiment).
Second, SG experiments highlight something even more peculiar: it
seems impossible to *know* the projection of the non-classical
momentum on different axes (*e.g.*, :math:`x` and :math:`z`) at the
same time. This prompts us to develop a new formalism (a *quantum
mechanical* formalism) to account for these behaviors. This leads
us to introduce a *quantum state vector* (noted :math:`\ket{\psi}`
and called *ket*) that includes all the properties of the
system. Students will remember that the new notation (called the
*Dirac* notation) should be a clear indication that we must adopt
a new approach, in strong departure from classical physics. We also
realize that the outcome of a measurement is *probabilistic* and we
introduce the *bra* state vector (noted :math:`\bra{\psi}`) that
allows us to calculate *probability amplitudes*
:math:`\braket{\psi}{\phi}`, which are complex numbers. The actual
*probability* is the square modulus of the probability amplitude
(that is: :math:`|\braket{\psi}{\phi}|^2`). Finally, we know that
the *ket* vectors live in the *Hilbert space*, with many properties
similar to that of other spaces encountered more generally when
learning linear algebra.
We also realize that if one considers all possible outcomes of the
experiment (*e.g.*, SGz where the projection on the :math:`z` axis is
measured), we can build a *complete set* of basis vectors, which
can be used to express any state vector (this is the principle of
*superposition*). Because the interpretation is probabilistic, we
are led to consider *expectation values* (noted
:math:`\expval{S_z}` for the measurement of the momentum projected
on the :math:`z` axis) and the *uncertainty*. It is important to
realize that even though the mathematical description looks like
that of a statistical analysis of large samples, the quantum
mechanical interpretation applies to *single* particles.
What are the Stern-Gerlach (SG) experiments?
--------------------------------------------
.. image:: _images/Chapter1/sterngerlach.png
This apparatus allows for the measurement of the magnetic moment of
a particle. The interaction between the magnetic moment and the
external magnetic field provides a net force on the particle. It
follows the deflection of the particle can be used to determine the
magnetic moment of the particle.
Important note: in order to generate a force, the magnetic field
must be *non-uniform* as the force goes as:
.. math::
F_{z}=\mu \cdot \frac{\partial \mathbf{B}}{\partial z} \simeq \mu_{z} \frac{\partial B_{z}}{\partial z}
It is essential to remember that in the SG experiments, the
particle does not have a classical angular momentum so any
deflection must be due to *something else*, which is not classical.
Through a series of SG experiments, one realizes that we must
define a new framework, beyond classical physics, to try to account
for those measurements.
Learning Material
-----------------
Copy of Slides
~~~~~~~~~~~~~~
.. raw:: html
The slides for Chapter 1 are available in pdf format here: 📂.
.. raw:: latex
The slides for Chapter 1 are available in pdf format \href{https://www.dropbox.com/s/ng5khct0tl59pl7/Chapter1.pdf}{here}.
Screencast
~~~~~~~~~~
.. raw:: html
.. raw:: latex
This lecture is available as a YouTube recording at this \href{https://www.youtube.com/embed/6LnEo1TW4Ck}{link}.
Key Learning Points
~~~~~~~~~~~~~~~~~~~
.. attention::
1. The ket vector is noted as :math:`\ket{\psi}`, it is an abstract
object. All the properties of the system it represents are
included in it but it does not mean we have easy access to that
knowledge.
2. The same information is included in the bra vector
:math:`\bra{\psi}`. This vector lives in the reciprocal space
and is introduced so that we can define the probability
amplitude :math:`\bra{\phi}\ket{\psi}` (also called a
*projection* or *dot product*), which is a complex number
providing the *probability amplitude* of finding state
:math:`\ket{\psi}` in state :math:`\ket{\phi}`.
3. The *probability* (a real number between 0 and 1) is given by
the square modulus of the probability amplitude:
.. math::
P=|\bra{\phi}\ket{\psi}|^2
This reads as ":math:`P` is the probability of finding state
:math:`\ket{\psi}` in state :math:`\ket{\phi}`". As a corollary,
we have: that a proper state vector is normalized:
.. math::
|\bra{\phi}\ket{\phi}|^2=1
4. The *labels* used to write the *ket* vectors indicate the
knowledge we have acquired about the state, *e.g.*,
:math:`\ket{+z}` represents states that are deflected upward
during the Stern-Gerlach experiment.
5. A *complete* basis is a set of ket vectors that can be used to
express any other vectors. For instance, the only two possible
outcomes of the Stern-Gerlach experiments are :math:`\ket{+z}`
and :math:`\ket{-z}`. This means all vectors can be expressed
using only those two vectors as the basis (this possibility is
also referred to as the *superposition
principle*). Mathematically, we translate this into:
.. math::
\ket{\psi}=c_+ \ket{+z} + c_- \ket{-z}
6. Those coefficients are **complex** numbers and are given by
*projections* on the basis vectors:
.. math::
|\psi\rangle=\underbrace{\langle+\mathbf{z} \mid \psi\rangle}_{c_{+}}|+\mathbf{z}\rangle+\underbrace{\langle-\mathbf{z} \mid \psi\rangle}_{c_{-}}|-\mathbf{z}\rangle
7. An operator, in quantum mechanics is noted as :math:`\hat{A}`
(that is: with a hat). It is a mathematical object that, when
applied to a state vector, yields another state vector.
8. The expectation value of an operator :math:`\hat{A}` (that is:
the average value you would measure when performing an
experiment that corresponds to the operator is noted
:math:`\expval{\hat{A}}`.
9. The uncertainty on the measurement of an operator :math:`\hat{S}_z` is
.. math::
\Delta S_{z}=\sqrt{\left\langle S_{z}^{2}\right\rangle-\left\langle S_{z}\right\rangle^{2}}
For instance, in the Stern-Gerlach experiment on a spin-1/2
particle, the uncertainly is :math:`~\hbar/2` since the average
value is zero and the only two possible outcomes are :math:`\pm
\hbar/2` with identical probability.
10. An important outcome of the SG experiment is that it is not
possible to know the value of the projection of the spin of
particle along :math:`x` and :math:`z` at the same time. This
can be seen by the fact that when using one outcome of the
:math:`SG_z` apparatus as input of an :math:`SG_x` apparatus
will always yield two possible outcomes with equal probability.
Test your knowledge
-------------------
1. Why does an electron get deflected during a Stern-Gerlach experiment?
A. because of the electric field generated by the magnet.
B. because of the presence of a uniform magnetic field and the fact the electrons have a magnetic moment.
C. because of the presence of a non-uniform magnetic field and the fact the electrons have a magnetic moment.
D. None of the other answers is correct.
2. What do we learn from the Stern-Gerlach (SG) experiments?
A. Electrons behave classically for the most part.
B. Electrons have an intrinsic magnetic moment even when their (classical) orbital momentum is zero.
C. An electron seems to have a quantum mechanical angular momentum, distinct from its orbital angular momentum and that momentum can take any continuous value between a maximum and a minimum.
3. What is a ket vector :math:`\ket{\psi}`?
A. It is a mathematical or abstract object that describes a given system. All of the information about the system is included in it, though one does not necessarily have easy access to that information.
B. It is a mathematical or abstract object that describes a given system. It includes some information about the system: the information one has already obtained with a measurement.
C. A ket vector is just a wave-function.
4. What is a bra vector :math:`\bra{\psi}`?
A. It is a mathematical object that has very little use outside of IQM.
B. It is a mathematical object that approximately represents the corresponding ket vector :math:`\ket{\psi}`.
C. It is a mathematical object that includes the exact same information on the system as the corresponding ket does. However, it "lives" in a different space and it allows IQM students to perform dot products.
5. In the context of SG experiments, how do you translate this in English: :math:`\braket{+z}{+x}`?
A. Smaller plus z vertical bar plus x larger.
B. Probability amplitude of finding a state :math:`\ket{+x}` in state :math:`\ket{+z}` after an SG :math:`_x` measurement.
C. Probability of finding a state :math:`\ket{+x}` in state :math:`\ket{+z}` after an SG :math:`_x` measurement.
6. A quantum state is defined up to an arbitrary phase factor. Said equivalently: mathematically the kets :math:`\ket{\psi}` and :math:`e^{i\delta}\ket{\psi}` (where :math:`\delta` is a real number) represent the same physical system.
A. True
B. False
7. When we say that :math:`\{ \ket{+x},\ket{-x}\}` represents a complete basis, what do we mean?
A. That the outcome of an SG :math:`_x` measurement can be any combination of :math:`\ket{+x}`, :math:`\ket{-x}`.
B. That any state :math:`\ket{\psi}` can be described as a superposition of :math:`\ket{+x}` and :math:`\ket{-x}`.
8. **What is expectation value and uncertainty of a measurement of SGz for this state:** :math:`\ket{\phi}=\frac{1}{\sqrt{3}}\ket{+z}+\sqrt{-\frac{2}{3}}\ket{-z}`?
9. **What is expectation value of a measurement of SGz for this state:** :math:`\ket{\phi}=\frac{1}{\sqrt{3}}\ket{+x}-\sqrt{\frac{2}{3}}\ket{-x}`?
(for this question, you can use the fact that :math:`\ket{\pm x}=\frac{1}{\sqrt{2}}(\ket{+z}\pm\ket{-z})`.
10. **Calculate expectation value and uncertainty for a measurement of SGZ for the following states:** :math:`\ket{\phi}=\frac{1}{\sqrt{4}}\ket{+x}-A\ket{-x}` (hint: first determine the value of :math:`A` to ensure that the state is a proper state).
11. :math:`\ket{\phi}=\frac{1}{\sqrt{3}}\ket{+y}-\sqrt{\frac{2}{3}}\ket{-y}`.
12. :math:`\ket{\phi}=\frac{1}{\sqrt{3}}\ket{+y}-A\ket{-x}` (hint: there is no typo in this question!).
.. hint::
Find the answer keys on this page: :ref:`answerkeys`. Don't cheat! Try solving the problems on your own first!
Homework Assignment
-------------------
Solve the following problems from the textbook: 1.3; 1.4; 1.5; 1.8, 1.11, 1.15