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 $\hat{S}_z$ 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:

$\hat{x}|x\rangle=x|x\rangle$

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

$|\psi\rangle=\int_{-\infty}^{\infty} d x|x\rangle\langle x \mid \psi\rangle$

This allows us to make the formal connection with the concept of wave-function:

$\langle x \mid \psi\rangle \equiv \psi(x)$

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:

$\hat{T}(a)|x\rangle=|x+a\rangle$

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 $dx$ :

$\hat{T}(d x)=1-\frac{i}{\hbar} \hat{p}_{x} d x$

The generator of translation turns out to be the linear momentum operator. It has very interesting properties, and we realize that

$\left[\hat{x}, \hat{p}_{x}\right]=i \hbar$

Next, we examine the time-evolution of the expectation value of position. This allows us to write the important Ehrenfest’s theorem:

$\begin{array}{ll} \frac{d\langle x\rangle}{d t}=\frac{\left\langle p_{x}\right\rangle}{m} & \frac{d\left\langle p_{x}\right\rangle}{d t}=\left\langle-\frac{d V}{d x}\right\rangle \end{array}$

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 $\lambda=\frac{h}{p}$ . 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 $\Delta x \Delta p_{x}=\hbar / 2$ .

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:

$\left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right]\langle x \mid E\rangle=E\langle x \mid E\rangle$

We realize that to solve this second-order different equation, one needs constraints on the wave-function and its derivative:

The wave-funciton must be continuous (so that the first derivative is well-defined), and

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:

$\sum_{i}\dyad{a_i}=1$

Orthogonality:

$\left\langle x_{j} \mid x_{i}\right\rangle=\delta_{i j}$

where we used the Kronecker delta.
Continuous basis:

$\int_{-\infty}^{\infty} d x|x\rangle\langle x|=1$

Orthogonality:

$\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right)$

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

The slides for Chapter 6 are available in pdf format here: 📂.

Screencast

Key Learning Points

Position operator: $\hat{x}|x\rangle=x|x\rangle$ . This is a Hermitian operator since all eigenvalues are real.
Completeness and orthogonality: $\int_{-\infty}^{\infty} d x|x\rangle\langle x|=1 \quad\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right)$
Probability of finding a particle between $x$ and $x+dx$ : $d x|\langle x \mid \psi\rangle|^{2}$
Wave-function: $\langle x \mid \psi\rangle \equiv \psi(x)$
Translation Operator: $\hat{T}(a)|x\rangle=|x+a\rangle$
Generator of translations: $\hat{T}(d x)=1-\frac{i}{\hbar} \hat{p}_{x} d x$
Commutation relationship: $\left[\hat{x}, \hat{p}_{x}\right]=i \hbar$ .
Ehrenfest’s theorem:

$\begin{array}{ll} \frac{d\langle x\rangle}{d t}=\frac{\left\langle p_{x}\right\rangle}{m} & \frac{d\left\langle p_{x}\right\rangle}{d t}=\left\langle-\frac{d V}{d x}\right\rangle \end{array}$
Representation of the linear momentum in the x basis:

$\hat{p}_{x} \underset{x \text { basis }}{\longrightarrow} \frac{\hbar}{i} \frac{\partial}{\partial x}$
Another identity relationship, using the eigen-states of $\hat{p}_x$ as a basis (this is known as completeness):

$\int d p \dyad{p} =1$

This basis is also orthogonal:

$\bra{p'}\ket{p}=\delta(p-p')$
We can also talk about a wave-function in momentum space: $\bra{p}\ket{\psi}\equiv\psi{p}$ . And we understand that $dp|$ bra{p}ket{psi}|^2` is the probability of finding the particle (described by $\ket{\psi}$ with a momentum between $p$ and $p+dp$ .
Using the position eigen-states as a basis, we find: $\bra{x}\ket{p}=\frac{1}{\sqrt{2 \pi \hbar}} e^{i p x / \hbar}$
Converting from the momentum-space to the position-space (and vice versa) is equivalent to a Fourier transform (and its inverse):

$\begin{array}{l} \langle p \mid \psi\rangle=\int d x\langle p \mid x\rangle\langle x \mid \psi\rangle=\int d x \frac{1}{\sqrt{2 \pi \hbar}} e^{-i p x / \hbar}\langle x \mid \psi\rangle \\ \langle x \mid \psi\rangle=\int d p\langle x \mid p\rangle\langle p \mid \psi\rangle=\int d p \frac{1}{\sqrt{2 \pi \hbar}} e^{i p x / \hbar}\langle p \mid \psi\rangle \end{array}$
Normalized Gaussian:

$f(x)=\frac{1}{\sqrt{\pi} a} e^{-x^{2} / a^{2}}$
Hamiltonian of a free particle (free means in the absence of a potential, and thus where the only term is the kinetic energy:

$\hat{H}=\frac{\hat{p}_{x}^{2}}{2 m}$

Test your knowledge

Write an expression of the identity operator using the eigen-states of the position operator $\hat{x}$ in one-dimension.
Write the completeness and orthogonality conditions of the continuous basis corresponding to the eigen-states of the momentum operator $\hat{p}$ .
Changing basis from an $x$ -representation to a $p_{x}$ -representation is mathematically equivalent to a…
Show that for the infinitesimal translation $|\psi\rangle \rightarrow\left|\psi^{\prime}\right\rangle=\hat{T}(\delta x)|\psi\rangle$ that $\langle x\rangle \rightarrow\langle x\rangle+\delta x \quad$ and $\quad\left\langle p_{x}\right\rangle \rightarrow\left\langle p_{x}\right\rangle$
The position operator, unlike the momentum operator, is Hermitian and has a continuous spectrum.
1. True
2. False
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.
1. True
2. False
The Kronecker and the Dirac deltas are just two names representing the same mathematical object.
1. True
2. False
The translation operator is Hermitian because its generator is unitary.
1. True
2. False
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 $\Psi(x, t)$ .
1. The continuity equation
2. The Fourier transform
3. Newton’s law
4. The Persans-Jackson law
5. The Schr"odinger equation
All in all, Ehrenfest’s theorem establishes an exact and one-to-one correspondence between Newtonian Physics and Quantum Mechanics. A. True B. False
For a free particle described by a Gaussian wave-packet at time $t=0$ : (a) this wave-function corresponds to the least uncertainty between position and momentum and (b) remains a Gaussian as it evolves in time.
1. 1. and (b) are true
2. 1. is true and (b) is false
3. 1. is false and (b) is true
4. 1. and (b) are false
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.
1. This is not true: quantum mechanics does not obey the rules of calculus
2. This is always true
3. This is often true, unless the potential is infinite (in this case (b) is not correct)
4. This is always false because I forgot to study for this quiz.

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. 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