.. _chapter6:
Chapter 6: Wave mechanics in one-dimension
++++++++++++++++++++++++++++++++++++++++++
.. note::
*The Universe is under no obligation to make sense to you* -- Neil deGrasse Tyson
Summary
-------
.. attention::
When dealing with the analysis of a quantum mechanical problem, it
is useful to make a clear formal distinction between a state vector
(which is an abstract object) and its representation. If the basis
used for the representation is *complete*, then the same
information is included in the representation and the abstract
object. The choice of representation is dictated by the specificity
of a problem. For instance, using the complete basis of the
eigen-states of :math:`\hat{S}_z` made sense when discussing SGz in
chapter 1. In this chapter, we introduce another type of
representation: the position-space representation. This is the most
popular representation and students are often introduced to quantum
mechanics directly using that represetnation rather than taking the
first step of focusing on the formal object itself that a state
vector constitutes.
In contrast to what we've seen so far, the position space
representation is a *continuous* basis. For instance, in
one-dimension, the basis vectors are the eigenkets of the *position
operator*:
.. math::
\hat{x}|x\rangle=x|x\rangle
The corresponding basis (which is complete since the position
operator is Hermitian) is a continuous basis (see box below for a
comparison between a continuous and a discrete basis).
Using the relationship above, we see that
.. math::
|\psi\rangle=\int_{-\infty}^{\infty} d x|x\rangle\langle x \mid \psi\rangle
This allows us to make the formal connection with the concept of *wave-function*:
.. math::
\langle x \mid \psi\rangle \equiv \psi(x)
In other words, a wave-function is nothing else than the projection
of the vector state on an eigenstate of the position operator. This
may seem strange as the wave-function is often introduced as the
starting point to students who first encounter quantum mechanics.
Next, after talking about the position operator, we look into the
translation operator, defined as the operator that moves an
eigenstate of the position operator by a given amount:
.. math::
\hat{T}(a)|x\rangle=|x+a\rangle
Of course, because it must preserve norms, the translation operator
is unitary and, in an almost Pavlovian reaction, we directly
suspect the existence of a *generator of translation*, by examining
the translation of an infinitesimal amount :math:`dx`:
.. math::
\hat{T}(d x)=1-\frac{i}{\hbar} \hat{p}_{x} d x
The generator of translation turns out to be the linear momentum
operator. It has very interesting properties, and we realize that
.. math::
\left[\hat{x}, \hat{p}_{x}\right]=i \hbar
Next, we examine the time-evolution of the expectation value of
position. This allows us to write the important Ehrenfest's
theorem:
.. math::
\begin{array}{ll}
\frac{d\langle x\rangle}{d t}=\frac{\left\langle p_{x}\right\rangle}{m} & \frac{d\left\langle p_{x}\right\rangle}{d t}=\left\langle-\frac{d V}{d x}\right\rangle
\end{array}
Here, we have to be careful! The equations look like classical
equations of motion. However, Ehrenfest's theorem does not claim
that the motion is essentially classical!
In this chapter, we have introduced an important operator: the
linear momentum (we, in fact, only introduce the *x* component but
generalizing is straightforward, as we shall see in future
chapters). Now, following the path we have adopted so far, we know
that this operator, which is Hermitian, has eigen-vectors that
constitute a complete basis! Similar to the position operator, the
basis is continuous (see box below). We studied the two new
(continuous) basis sets we have encountered in this chapter (see
*key learning points* below). In particular, we find that the de
Broglie relationship stem directly from the fact that the particle
has a wavelength (*i.e.*, periodicity) given by
:math:`\lambda=\frac{h}{p}`. We also find an interesting result:
comparing the orthogonality (point #11 below) and the position
space representation of the momentum state (point #12 below)
indicates that the momentum state is a sharp spike in momentum
space but is an extended wave in position space. When dealing with
this *duality*, students will remember that the two representations
are equivalent but are expressed from two different points of
view. Finally, the relationship between the two representations
turns out to be nothing else than the Fourier transform
encountered in other fields.
One particularly interesting state is the one described by a
Gaussian wave-packet. The use of the name *wave-packet* is easy to
understand: that state is a *packet* of eigen-states of the
momentum (or position) operators! One reason why physicists love
Gaussians is because they are easy to deal with mathematically. In
addition, a Gaussian has the particularity of also having a
Gaussian as its Fourier transform. However, remembering the
uncertainty relationship, we also know that if the Gaussian is
broad in the momentum-space, it is very sharp in position-space
(and *vice versa*). In fact, a Gaussian wave-packet is the minimum
uncertainty state, for which :math:`\Delta x \Delta p_{x}=\hbar /
2`.
We continue the chapter with the time-evolution of a free particle
initially described by a Gaussian wave-packet. We realize that the
wave-packet spreads (in space) as it evolves. This is not
surprising as the state is a superposition of a variety of momentum
eigen-states and thus not all components evolve at the same rate!
We also remember that while the state may look like a Gaussian as
time evolves, it is not a Gaussian anymore (in fact, the
uncertainty of the momentum is a constant while the state spreads,
since a momentum state remains an eigenstate of the free-particle
Hamiltonian at all times).
The uncertainty relationship is key to understanding the
double-slit experiment and we particularly discuss that if one
knows which slit a particle went through, then no interference
pattern can be measured.
We finally make the connection between this course and quantum
physics courses you took as sophomores and we write the
time-independent Schrodinger's equation in position space:
.. math::
\left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right]\langle x \mid E\rangle=E\langle x \mid E\rangle
We realize that to solve this second-order different equation, one
needs constraints on the wave-function and its derivative:
1) The wave-funciton must be continuous (so that the first
derivative is well-defined), and
2) The first derivative must also be continuous everywhere
**provided the potential is finite**.
This allows us to solve a number of 1D problems, such as the
particle in the box problem and the scattering by a potential
problem.
.. note::
**Completeness relation: discrete versus continuous**
* Discrete basis:
.. math::
\sum_{i}\dyad{a_i}=1
Orthogonality:
.. math::
\left\langle x_{j} \mid x_{i}\right\rangle=\delta_{i j}
where we used the Kronecker delta.
* Continuous basis:
.. math::
\int_{-\infty}^{\infty} d x|x\rangle\langle x|=1
Orthogonality:
.. math::
\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right)
where we used the Dirac delta.
**Hint:** when you try to prove something in quantum mechanics,
remember you can insert "identity" operators anywhere you
want. Very often, the relationships above can make your life very
easy!
Learning Material
-----------------
Copy of Slides
~~~~~~~~~~~~~~
.. raw:: html
The slides for Chapter 6 are available in pdf format here: <a href="https://www.dropbox.com/s/6h94gn4vmwf2mhc/Chapter6.pdf?dl=0">📂</a>.
.. raw:: latex
The slides for Chapter 6 are available in pdf format \href{https://www.dropbox.com/s/6h94gn4vmwf2mhc/Chapter6.pdf}{here}.
Screencast
~~~~~~~~~~
.. raw:: html
<iframe width="560" height="315" src="https://www.youtube.com/embed//oBupvq_iyEE" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
.. raw:: latex
This lecture is available as a YouTube recording at this \href{https://www.youtube.com/embed//oBupvq_iyEE}{link}.
.. admonition:: Key Learning Points
1. Position operator: :math:`\hat{x}|x\rangle=x|x\rangle`. This is a Hermitian operator since all eigenvalues are real.
2. Completeness and orthogonality: :math:`\int_{-\infty}^{\infty} d x|x\rangle\langle x|=1 \quad\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right)`
3. Probability of finding a particle between :math:`x` and :math:`x+dx`: :math:`d x|\langle x \mid \psi\rangle|^{2}`
4. Wave-function: :math:`\langle x \mid \psi\rangle \equiv \psi(x)`
5. Translation Operator: :math:`\hat{T}(a)|x\rangle=|x+a\rangle`
6. Generator of translations: :math:`\hat{T}(d x)=1-\frac{i}{\hbar} \hat{p}_{x} d x`
7. Commutation relationship: :math:`\left[\hat{x}, \hat{p}_{x}\right]=i \hbar`.
8. Ehrenfest's theorem:
.. math::
\begin{array}{ll}
\frac{d\langle x\rangle}{d t}=\frac{\left\langle p_{x}\right\rangle}{m} & \frac{d\left\langle p_{x}\right\rangle}{d t}=\left\langle-\frac{d V}{d x}\right\rangle
\end{array}
9. Representation of the linear momentum in the *x* basis:
.. math::
\hat{p}_{x} \underset{x \text { basis }}{\longrightarrow} \frac{\hbar}{i} \frac{\partial}{\partial x}
10. Another identity relationship, using the eigen-states of :math:`\hat{p}_x` as a basis (this is known as *completeness*):
.. math::
\int d p \dyad{p} =1
This basis is also orthogonal:
.. math::
\bra{p'}\ket{p}=\delta(p-p')
11. We can also talk about a *wave-function in momentum space*:
:math:`\bra{p}\ket{\psi}\equiv\psi{p}`. And we understand that
:math:`dp|`\bra{p}\ket{\psi}|^2` is the probability of finding
the particle (described by :math:`\ket{\psi}` with a momentum
between :math:`p` and :math:`p+dp`.
12. Using the position eigen-states as a basis, we find:
:math:`\bra{x}\ket{p}=\frac{1}{\sqrt{2 \pi \hbar}} e^{i p x /
\hbar}`
13. Converting from the momentum-space to the position-space (and
*vice versa*) is equivalent to a Fourier transform (and its
inverse):
.. math::
\begin{array}{l}
\langle p \mid \psi\rangle=\int d x\langle p \mid x\rangle\langle x \mid \psi\rangle=\int d x \frac{1}{\sqrt{2 \pi \hbar}} e^{-i p x / \hbar}\langle x \mid \psi\rangle \\
\langle x \mid \psi\rangle=\int d p\langle x \mid p\rangle\langle p \mid \psi\rangle=\int d p \frac{1}{\sqrt{2 \pi \hbar}} e^{i p x / \hbar}\langle p \mid \psi\rangle
\end{array}
14. Normalized Gaussian:
.. math::
f(x)=\frac{1}{\sqrt{\pi} a} e^{-x^{2} / a^{2}}
15. Hamiltonian of a free particle (free means *in the absence of a
potential*, and thus where the only term is the kinetic energy:
.. math::
\hat{H}=\frac{\hat{p}_{x}^{2}}{2 m}
Test your knowledge
-------------------
1. Write an expression of the identity operator using the eigen-states of the position operator :math:`\hat{x}` in one-dimension.
2. Write the completeness and orthogonality conditions of the continuous basis corresponding to the eigen-states of the momentum operator :math:`\hat{p}`.
3. Changing basis from an :math:`x` -representation to a :math:`p_{x}` -representation is mathematically equivalent to a...
4. Show that for the infinitesimal translation :math:`|\psi\rangle \rightarrow\left|\psi^{\prime}\right\rangle=\hat{T}(\delta x)|\psi\rangle` that :math:`\langle x\rangle \rightarrow\langle x\rangle+\delta x \quad` and :math:`\quad\left\langle p_{x}\right\rangle \rightarrow\left\langle p_{x}\right\rangle`
5. The position operator, unlike the momentum operator, is Hermitian and has a continuous spectrum.
A. True
B. False
6. The origin of the uncertainty principle between the position of a particle and its linear momentum can be traced back to the fact that the position and momentum operators do not commute.
A. True
B. False
7. The Kronecker and the Dirac deltas are just two names representing the same mathematical object.
A. True
B. False
8. The translation operator is Hermitian because its generator is unitary.
A. True
B. False
9. Among the answers below, which ones fits best the following definition? An equation that must hold in order for the non-relativistic Hamiltonian operator to yield an energy expectation value for a wave function :math:`\Psi(x, t)`.
A. The continuity equation
B. The Fourier transform
C. Newton's law
D. The Persans-Jackson law
E. The Schr\"odinger equation
10. All in all, Ehrenfest's theorem establishes an exact and one-to-one correspondence between Newtonian Physics and Quantum Mechanics.
A. True
B. False
11. For a free particle described by a Gaussian wave-packet at time :math:`t=0`: (a) this wave-function corresponds to the least uncertainty between position and momentum and (b) remains a Gaussian as it evolves in time.
A. (a) and (b) are true
B. (a) is true and (b) is false
C. (a) is false and (b) is true
D. (a) and (b) are false
12. When solving time-independent Schroedinger equation, we must make sure that (a) the solution is continuous everywhere and (b) its derivative is continuous everywhere as well.
A. This is not true: quantum mechanics does not obey the rules of calculus
B. This is always true
C. This is often true, unless the potential is infinite (in this case (b) is not correct)
D. This is always false because I forgot to study for this quiz.
.. 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: 6.1, 6.2, 6.6, 6.9, 6.13
Recitation Assignment
---------------------
Solve the following problems from the textbook: 6.4, 6.5, 6.17