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📌How to Derive the Hamiltonian for Phonons in a Crystal Lattice

Understanding the dynamics of atoms in a crystal lattice is crucial for exploring material properties such as thermal and electrical conductivity, elasticity, and more. This post will walk you through the derivation of the Hamiltonian for phonons—quantized sound waves or vibrations in a crystal lattice—using a step-by-step approach. Let’s dive into the mathematical framework that allows us to describe these vibrations in terms of phonon modes using Fourier transforms.

1. Starting with Classical Dynamics

In a crystal, atoms are arranged in a repeating pattern and can oscillate around their equilibrium positions. These oscillations can be modeled using classical mechanics principles.

Classical Kinetic Energy: For a lattice with atoms of mass mmm, the kinetic energy is given by: $\boldsymbol{Kinetic Energy = \frac{1}{2m} \sum_{n} p_n^2}$ . Here, $\boldsymbol{p_n}$ is the momentum of the n-th atom.

Classical Potential Energy: For nearest-neighbor interactions, the potential energy, assuming a spring-like interaction, is: $\text{Potential Energy} = \frac{1}{2} \sum_{n} K (u_{n+1} - u_n)^2$ , where K is the spring constant between neighboring atoms.

2. Transitioning to Reciprocal Space (Fourier Transform)

To simplify the problem, we use Fourier transforms to express both displacement and momentum: $\boldsymbol{u_n = \sum_k A_k e^{ikna}}$ and $\boldsymbol{p_n = \sum_k P_k e^{ikna}}$ , where $\boldsymbol{A_k}$ and $\boldsymbol{P_k}$ are the Fourier components of displacement and momentum, respectively, and \latex \boldsymbol{a}$ is the lattice constant.

3. Expressing Energy in k-space

Kinetic Energy in k-space: By substituting the Fourier expansions into the kinetic energy expression and using the orthogonality of the exponential terms, we get: $\text{Kinetic Energy} = \frac{N}{2m} \sum_k P_k P_{-k}$ where N is the total number of lattice sites.

Potential Energy in k-space: Similarly, substituting into the potential energy and simplifying using the properties of exponential sums leads to: $\text{Potential Energy} = \frac{N}{2} \sum_k 2K \sin^2\left(\frac{ka}{2}\right) A_k A_{-k}$ .

4. Constructing the Hamiltonian

Combining these energy expressions, the Hamiltonian for the system in k-space becomes: $H = \frac{1}{2} \sum_k \left(\frac{N}{m} P_k P_{-k} + m \omega_k^2 A_k A_{-k} \right)$ , where: $\omega_k = \sqrt{\frac{2K}{m} \sin^2\left(\frac{ka}{2}\right)}$ . This expression encapsulates the total energy (kinetic plus potential) of the lattice vibrations.

5. Quantum Mechanical Formalism

In quantum mechanics, we promote the $A_k, A_{-k}$ , and $P_kP_{-k}$ to operators and introduce phonon creation and annihilation operators to quantize the Hamiltonian. This framework allows us to study the properties of phonons and their interactions within the crystal.

Conclusion

This derivation illustrates how classical mechanics concepts, combined with Fourier analysis and quantum mechanics, provide a comprehensive description of lattice dynamics. The resulting Hamiltonian for phonons enables physicists to predict various material properties and understand deeper phenomena in solid-state physics.

This blog post is intended for educational purposes, offering a step-by-step approach to understanding complex physical concepts in material science and condensed matter physics. Whether you are a student or a researcher, mastering these principles is crucial for further exploration in the field.

Detailed derivation here:

Step 1: Expressing Kinetic Energy in k-space

Classical Kinetic Energy:

The kinetic energy of a lattice in real space is given by: $\text{Kinetic Energy} = \frac{1}{2m} \sum_{n} p_n^2$

where $p_n$ is the momentum of the n-th atom. Transforming this momentum to k-space: $p_n = \sum_k P_k e^{ikna}$ .

Substitute into Kinetic Energy Expression:

$\text{Kinetic Energy} = \frac{1}{2m} \sum_{n} \left(\sum_k P_k e^{ikna}\right) \left(\sum_{k'} P_{k'} e^{ik'na}\right)$

Using Orthogonality of Exponential Terms:

We employ the orthogonality of exponential terms to simplify the kinetic energy expression, leveraging the properties of the discrete Fourier transform and the periodic boundary conditions of the lattice:

$\sum_{n} e^{i(k+k')na} = N \delta_{k,-k'}$

where $\delta_{k,-k'}$ is the Kronecker delta, which is non-zero only when $k' = -k$ . This simplification leverages the periodic boundary conditions and the properties of the discrete Fourier transform. Substituting back, we get: $\text{Kinetic Energy} = \frac{N}{2m} \sum_k P_k P_{-k}$

Step 2: Expressing Potential Energy in k-space

Classical Potential Energy:

The potential energy in a linear chain with nearest-neighbor interactions can be described as:

This equation accounts for the energy due to the displacement differences between adjacent atoms in the chain, where K is the spring constant.

Transforming Displacements into k-space

To simplify the analysis and to handle complex systems more efficiently, we use Fourier transforms to convert the displacements from real space to reciprocal space (k-space). The displacement of the n-th atom and its next neighbor in the chain are represented as follows:

Potential Energy in k-space

Substituting these Fourier transformed displacements into the potential energy expression, we get:

This expands to:

Simplifying the Expression Using Orthogonality

To simplify further, apply the orthogonality properties of the exponential functions:

Using the identity:

This reduces the expression for potential energy in k-space:

The final formula correctly represents the potential energy due to lattice vibrations analyzed in k-space, offering a framework for examining dynamics and thermodynamic properties of crystal lattices in solid-state physics.

Substituting Displacements into the Potential Energy Expression

After transforming the displacements into k-space, substitute these into the potential energy equation for nearest-neighbor interactions. The transformation and subsequent squaring of the difference between adjacent displacements yield:

Transformed Potential Energy Expression:

This expands to:

Breaking Down the Expanded Expression

Expanding the squared term and applying the Fourier transform properties, the potential energy expression becomes:

Simplifying Using Orthogonality of Exponential Functions

Utilize the orthogonality property of the exponential terms, where the sum over all lattice points n introduces a Kronecker delta $\delta_{k,k'}$ , simplifying the double summation over wave vectors k and k′ by collapsing it when $k \neq k'$ :

Thus, the potential energy in k-space simplifies to:

This is further simplified using the identity $|e^{ika} - 1|^2 = 2 - 2\cos(ka)|$ :

Final Expression for Potential Energy in k-space

The final expression for potential energy, reflecting the contributions of phonon modes characterized by their wave vectors in the crystal lattice, is:

This formatted sequence provides a clear step-by-step transformation of the potential energy from real space to reciprocal space, highlighting the mathematical manipulation and simplifications used in solid-state physics to handle complex crystal lattice dynamics effectively.

Using Orthogonality of Exponential Terms:

Start by applying the orthogonality property of the exponential terms, which simplifies the expression significantly:

This relationship indicates that the sum over all sites n is non-zero only when k equals k′′, which is a fundamental property used in Fourier analysis, simplifying the sums over wave vectors.

Transforming Potential Energy into k-space:

With this simplification, the potential energy in the crystal lattice, which was initially expressed in terms of real-space displacements, can now be fully expressed in k-space:

Breaking Down the Potential Energy Formula:

The term $|e^{ika} - 1|^2|$ can be expanded using Euler’s formula, where $e^{ix} = \cos(x) + i\sin(x)$ , thus:

Therefore, substituting back into the potential energy expression gives:

Final Expression for Potential Energy in k-space:

Simplifying further using the trigonometric identity for cosine, the potential energy expression becomes:

This final expression reflects the potential energy contribution from all phonon modes in the lattice, characterized by their wave vector k, and encapsulates the effects of lattice vibrations in terms of their amplitude $A_k$ and the wave vector-dependent sinusoidal term.

This method of transforming and simplifying potential energy into k-space is crucial for analyzing the dynamics and thermodynamic properties of materials in the field of solid-state physics. The use of k-space representations allows physicists to handle complex interactions within the crystal lattice effectively, paving the way for deeper understanding and innovative material science applications.

Step 3: The Hamiltonian in k-space

After obtaining the expressions for kinetic and potential energies in terms of the Fourier components of the displacements and momenta, we can construct the Hamiltonian for the system. The Hamiltonian, which represents the total energy (kinetic plus potential) of the system, is a crucial tool in quantum mechanics and solid state physics for studying the dynamics of lattice vibrations.

Hamiltonian Expression:

The Hamiltonian in k-space is given by:

This expression combines the transformed kinetic and potential energy terms:

Kinetic Energy Term: represents the kinetic energy in kkk-space, where PkP_kPk and P−kP_{-k}P−k are the momentum components for wavevector kkk and its negative, reflecting the preservation of momentum in the lattice.
Potential Energy Term: encapsulates the potential energy due to displacements from equilibrium positions, characterized by the sinusoidal square term that varies with the wavevector kkk.

Simplified Hamiltonian with Angular Frequency:

We can introduce the angular frequency $\omega_k$ for the phonon modes to simplify the expression:

Thus, the Hamiltonian simplifies to:

This form of the Hamiltonian clearly indicates the relationship between the wavevector-dependent angular frequency $\omega_k$ and the kinetic and potential energies of the system, providing a comprehensive description of the energy distribution among the phonon modes in the lattice.

Conclusion

The Hamiltonian in k-space effectively describes how the total energy of the crystal lattice is distributed among various phonon modes, each characterized by a distinct wavevector k. This framework is essential for exploring physical phenomena such as heat capacity, thermal conductivity, and other dynamic properties of materials at the microscopic level. By understanding these principles, researchers can predict material behavior under various conditions, facilitating advancements in material science and solid-state physics.

This derivation combines classical lattice dynamics with the properties of Fourier transforms under periodic boundary conditions, allowing the expressions for kinetic and potential energy in terms of momentum and displacement in kkk-space to be derived step by step.

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Deciphering Uncertainty: Jacob Bernoulli’s Legacy in Probability Theory

7 Mar

In this series, our primary aim is to demystify and popularize intricate mathematical structures, making them accessible to the everyday individual, both aspirational learners and mindful engineers. Even though mathematics is seen as mysterious and abstract, it is full of deep symmetry, profound truths, and beautiful things. Gaining mathematics literacy improves our ability to solve problems and creates new opportunities across many industries. A mathematically literate society is more powerful and capable of making the most use of equations and numbers to improve living standards and get a better understanding of the universe.

The 17th-century Swiss mathematician Jacob Bernoulli is commonly referred to as the “Father of Probability” because of the fundamental contributions he made to the science of probability and to mathematics as a whole. The moniker “Father of Uncertainty Quantification” may not be used often, but it accurately captures Bernoulli’s contribution to the creation of mathematical instruments for managing and comprehending uncertainty.

Contributions to Uncertainty Quantification:

Law of Large Numbers: In his posthumously released book “Ars Conjectandi” in 1713, Jacob Bernoulli made the most important contribution to the area of probability and uncertainty quantification: he formulated the Law of Large Numbers. This theorem ensures that, as the number of trials rises, the average of the outcomes becomes closer to the predicted value, hence lowering uncertainty in predictions. It is fundamental to understanding the behaviour of averages of random variables.
Binary Systems and Bernoulli Trials: He investigated binary systems, or what are now known as Bernoulli trials, in which there are only two potential outcomes (such as tossing a coin). The foundation for comprehending random processes and estimating the probability of various outcomes was established by this study, which is essential for calculating uncertainty in systems with probabilistic rather than deterministic outcomes.
Combination Theory: Combination theory, which is essential for computing probabilities and comprehending the distribution of outcomes in complicated systems, saw major advancements under Bernoulli’s leadership. This work is crucial for industries like banking, insurance, and engineering that deal with risk and uncertainty.
Risk Management: Bernoulli’s contributions to probability theory gave rise to the instruments that actuarial science and risk management use today. His research makes it possible to evaluate and quantify risks in ambiguous situations, which improves decision-making in these situations.
Psychology of Decision Making: In addition, Bernoulli discussed the psychological implications of making decisions in the face of uncertainty, which paved the way for the eventual exploration of the Bernoulli principle, commonly known as utility theory. This principle introduces the idea of individual risk tolerance into the measurement of uncertainty by recommending that decision-makers compare the projected utility values of hazardous and uncertain prospects in order to make their selection.

Impact on Uncertainty Quantification:

Numerous fields that deal with uncertainty have been profoundly and permanently impacted by the work of Jacob Bernoulli on probability theory. His mathematical formulas establish the foundation for contemporary statistics, risk assessment, and decision sciences by offering a means of modelling, analysing, and making predictions about systems with uncertain outcomes. A key figure in the history of uncertainty quantification, Bernoulli’s concepts have enabled scientists, engineers, economists, and decision-makers to use a methodical and scientific approach to managing the unknown.

The Law of Large Numbers is Jacob Bernoulli’s most famous contribution to the understanding and measurement of uncertainty, even though he did not formulate it as an equation like we would do today. Rather, his work established the basic ideas that later mathematicians would formalise into mathematical equations. The Law of Large Numbers essentially deals with the outcomes of conducting an experiment many times. It states that the average of the results from many trials should be near to the expected value, and that the average of the results should tend to approach the expected value as more trials are conducted.

Law of Large Numbers (Conceptual Explanation)

While Bernoulli did not provide a modern equation for this law, the essence of his discovery can be expressed in a simplified modern notation as follows: Given a large number of trials n, the average result $[\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i]$ of these trials for a random variable X with expected value $latex E(X)=μ$, the Law of Large Numbers tells us that:

$\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \rightarrow \mu \quad \text{as} \quad n \rightarrow \infty$

Examples Illustrating Bernoulli’s Concept of Uncertainty

Example 1: Coin Tossing

Consider the act of tossing a fair coin. Each toss has two possible outcomes: heads or tails. If we define success as landing on heads, the expected probability of success p in each trial is 0.5.

If you toss the coin a large number of times, say n=1000, the Law of Large Numbers suggests that the proportion of heads ( $[\bar{X}_{1000} = \frac{1}{1000} \sum_{i=1}^{1000} X_i]$ ) will be very close to 0.5. The more you toss the coin, the closer the proportion will get to the expected value of 0.5.

Example 2: Rolling a Die

Imagine rolling a fair six-sided die. The expected value E(X) of the outcome is 3.5, since each side has an equal probability of landing face up, and the average (mean) of all possible outcomes (1 through 6) is 3.5.

If you roll the die a large number of times, the average value of the results $[\bar{X}_{n} = \frac{1}{n} \sum_{i=1}^{n} X_i]$ should approach 3.5, demonstrating the Law of Large Numbers. For instance, if n=10000 rolls, the average result should be very close to 3.5.

Bernoulli’s Trials

Another significant concept introduced by Bernoulli is the Bernoulli trial, which is a random experiment where there are only two possible outcomes (e.g., success or failure). The probability of success in each trial is p and the probability of failure is (1−p). Though Bernoulli’s Trials themselves do not directly describe uncertainty, they form the basis of many statistical methods for dealing with probabilistic events.

Bernoulli’s Equation for a Single Trial’s Expected Value and Variance

For a Bernoulli trial with success probability p, the expected value E(X) and variance Var(X) are given by:

$E(X)=p$
Var(X)=p (1−p)

These fundamental concepts and the mathematical framework Bernoulli developed form the basis of probability theory and the quantification of uncertainty, influencing countless applications in science, engineering, economics, and decision theory.

Application in the Stock Market:

Expected Return and Variance in Portfolio Theory:

Modern portfolio theory, which evaluates how investors may optimise their portfolios through diversification to maximise anticipated return for a given level of risk, is based on Bernoulli’s insights into probability and risk.

Expected Return of a Portfolio: The computation of a portfolio’s anticipated return involves averaging the projected returns of each individual item in the portfolio, with the weights indicating the relative importance of each asset:

$E(R_p)=\sum_{i=1}^n w_i E(R_i)$ .

Portfolio Variance: In addition to accounting for the variances of individual assets, the variance of the portfolio’s return, which serves as a gauge of risk, also considers the covariance between pairs of assets:

$Var(R_p)=\sum_{i=1}^n\sum_{j=1}^nw_iw_jCov(R_i,R_j)$ ,

where $Cov(R_i,R_j)$ is the covariance between the returns of assets i and j; $E(R_p)$ is the expected return of the portfolio; $w_i$ is the weight of the ith asset in the portfolio; $E(R_i)$ is the expected return of the ith asset; n is the number of assets in the portfolio.

By utilising Bernoulli’s Law of Large Numbers and his seminal work on probability, investors may better comprehend and control the inherent unpredictability of financial markets.

Application in Engineering:

Reliability Engineering and Risk Assessment:

In engineering, failure rates and component dependability are predicted using statistical techniques and the Law of Large Numbers.

Failure Rate (λ) Calculation: In reliability engineering, the failure rate (λ) of a component is often estimated from historical failure data, $\lambda=\frac{Total operational time}{Number of failures}$
System Reliability: For a series system of n independent components, the system reliability (Rs) can be calculated as the product of the reliabilities of individual components: $Rs=\Pi_{i=1}^n Ri$ ,where Ri is the reliability of component i.
Application in Quality Control (Six Sigma):The Six Sigma methodology in quality control uses statistical methods to reduce defects and variability in manufacturing processes. It aims to have processes operate within a certain range of standard deviations (σ) from the mean to minimize the defect rate. The process capability index (Cp) is a measure used in Six Sigma to quantify how well a process fits within its specification limits:
$Cp=\frac{Specification Upper Limit-Specification Lower Limit}{6 \sigma}$ :
σ is the standard deviation of the process output.

These instances highlight how Bernoulli’s theories of probability and statistical analysis are applied directly to the engineering and stock market domains, managing uncertainty and maximising results.