Software

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Software

 

A diagram showing how the user interacts with application software on a typical desktop computer. The application software layer interfaces with the operating system, which in turn communicates with the hardware. The arrows indicate information flow.

Software is a set of computer programs and associated documentation and data. This is in contrast to hardware, from which the system is built and which actually performs the work.

At the lowest programming level, executable code consists of machine language instructions supported by an individual processor—typically a central processing unit (CPU) or a graphics processing unit (GPU). Machine language consists of groups of binary values signifying processor instructions that change the state of the computer from its preceding state. For example, an instruction may change the value stored in a particular storage location in the computer—an effect that is not directly observable to the user. An instruction may also invoke one of many input or output operations, for example, displaying some text on a computer screen, causing state changes that should be visible to the user. The processor executes the instructions in the order they are provided, unless it is instructed to "jump" to a different instruction or is interrupted by the operating system. As of 2023, most personal computers, smartphone devices, and servers have processors with multiple execution units, or multiple processors performing computation together, so computing has become a much more concurrent activity than in the past.

The majority of software is written in high-level programming languages. They are easier and more efficient for programmers because they are closer to natural languages than machine languages. High-level languages are translated into machine language using a compiler, an interpreter, or a combination of the two. Software may also be written in a low-level assembly language that has a strong correspondence to the computer's machine language instructions and is translated into machine language using an assembler.

History

Main article: History of software

An algorithm for what would have been the first piece of software was written by Ada Lovelace in the 19th century, for the planned Analytical Engine. She created proofs to show how the engine would calculate Bernoulli numbers. Because of the proofs and the algorithm, she is considered the first computer programmer.

The first theory about software, prior to the creation of computers as we know them today, was proposed by Alan Turing in his 1936 essay, On Computable Numbers, with an Application to the Entscheidungsproblem (decision problem). This eventually led to the creation of the academic fields of computer science and software engineering; both fields study software and its creation. Computer science is the theoretical study of computer and software (Turing's essay is an example of computer science), whereas software engineering is the application of engineering principles to development of software.

In 2000, Fred Shapiro, a librarian at the Yale Law School, published a letter revealing that John Wilder Tukey's 1958 paper "The Teaching of Concrete Mathematics" contained the earliest known usage of the term "software" found in a search of JSTOR's electronic archives, predating the Oxford English Dictionary's citation by two years. This led many to credit Tukey with coining the term, particularly in obituaries published that same year, although Tukey never claimed credit for any such coinage. In 1995, Paul Niquette claimed he had originally coined the term in October 1953, although he could not find any documents supporting his claim. The earliest known publication of the term "software" in an engineering context was in August 1953 by Richard R. Carhart, in a Rand Corporation Research Memorandum.

Types

See also: Software categories

 

Blender, a free software program

On virtually all computer platforms, software can be grouped into a few broad categories.

Purpose, or domain of use

Based on the goal, computer software can be divided into:

• Application software uses the computer system to perform special functions beyond the basic operation of the computer itself. There are many different types of application software because the range of tasks that can be performed with a modern computer is so large—see list of software.
• System software manages hardware behaviour, as to provide basic functionalities that are required by users, or for other software to run properly, if at all. System software is also designed for providing a platform for running application software, and it includes the following:
  • Operating systems are essential collections of software that manage resources and provide common services for other software that runs "on top" of them. Supervisory programs, boot loaders, shells and window systems are core parts of operating systems. In practice, an operating system comes bundled with additional software (including application software) so that a user can potentially do some work with a computer that only has one operating system.
  • Device drivers operate or control a particular type of device that is attached to a computer. Each device needs at least one corresponding device driver; because a computer typically has at minimum at least one input device and at least one output device, a computer typically needs more than one device driver.
  • Utilities are computer programs designed to assist users in the maintenance and care of their computers.
• Malicious software, or malware, is software that is developed to harm or disrupt computers. Malware is closely associated with computer-related crimes, though some malicious programs may have been designed as practical jokes.

Nature or domain of execution

• Desktop applications such as web browsers and Microsoft Office and LibreOffice and WordPerfect, as well as smartphone and tablet applications (called "apps").
• JavaScript scripts are pieces of software traditionally embedded in web pages that are run directly inside the web browser when a web page is loaded without the need for a web browser plugin. Software written in other programming languages can also be run within the web browser if the software is either translated into JavaScript, or if a web browser plugin that supports that language is installed; the most common example of the latter is ActionScript scripts, which are supported by the Adobe Flash plugin.
• Server software, including:
  • Web applications, which usually run on the web server and output dynamically generated web pages to web browsers, using e.g. PHP, Java, ASP.NET, or even JavaScript that runs on the server. In modern times these commonly include some JavaScript to be run in the web browser as well, in which case they typically run partly on the server, partly in the web browser.
• Plugins and extensions are software that extends or modifies the functionality of another piece of software, and require that software be used in order to function.
• Embedded software resides as firmware within embedded systems, devices dedicated to a single use or a few uses such as cars and televisions (although some embedded devices such as wireless chipsets can themselves be part of an ordinary, non-embedded computer system such as a PC or smartphone). In the embedded system context there is sometimes no clear distinction between the system software and the application software. However, some embedded systems run embedded operating systems, and these systems do retain the distinction between system software and application software (although typically there will only be one, fixed application which is always run).
• Microcode is a special, relatively obscure type of embedded software which tells the processor itself how to execute machine code, so it is actually a lower level than machine code. It is typically proprietary to the processor manufacturer, and any necessary correctional microcode software updates are supplied by them to users (which is much cheaper than shipping replacement processor hardware). Thus an ordinary programmer would not expect to ever have to deal with it.

Programming tools

Main article: Programming tool

Programming tools are also software in the form of programs or applications that developers use to create, debug, maintain, or otherwise support software.

Software is written in one or more programming languages; there are many programming languages in existence, and each has at least one implementation, each of which consists of its own set of programming tools. These tools may be relatively self-contained programs such as compilers, debuggers, interpreters, linkers, and text editors, that can be combined to accomplish a task; or they may form an integrated development environment (IDE), which combines much or all of the functionality of such self-contained tools. IDEs may do this by either invoking the relevant individual tools or by re-implementing their functionality in a new way. An IDE can make it easier to do specific tasks, such as searching in files in a particular project. Many programming language implementations provide the option of using both individual tools or an IDE.

Topics

Architecture

See also: Software architecture

People who use modern general purpose computers (as opposed to embedded systems, analog computers and supercomputers) usually see three layers of software performing a variety of tasks: platform, application, and user software.

• Platform software: The platform includes the firmware, device drivers, an operating system, and typically a graphical user interface which, in total, allow a user to interact with the computer and its peripherals (associated equipment). Platform software often comes bundled with the computer. On a PC one will usually have the ability to change the platform software.
• Application software: Application software is what most people think of when they think of software. Typical examples include office suites and video games. Application software is often purchased separately from computer hardware. Sometimes applications are bundled with the computer, but that does not change the fact that they run as independent applications. Applications are usually independent programs from the operating system, though they are often tailored for specific platforms. Most users think of compilers, databases, and other "system software" as applications.
• User-written software: End-user development tailors systems to meet users' specific needs. User software includes spreadsheet templates and word processor templates. Even email filters are a kind of user software. Users create this software themselves and often overlook how important it is. Depending on how competently the user-written software has been integrated into default application packages, many users may not be aware of the distinction between the original packages, and what has been added by co-workers.

Execution

Main article: Execution (computing)

Computer software has to be "loaded" into the computer's storage (such as the hard drive or memory). Once the software has loaded, the computer is able to execute the software. This involves passing instructions from the application software, through the system software, to the hardware which ultimately receives the instruction as machine code. Each instruction causes the computer to carry out an operation—moving data, carrying out a computation, or altering the control flow of instructions.

Data movement is typically from one place in memory to another. Sometimes it involves moving data between memory and registers which enable high-speed data access in the CPU. Moving data, especially large amounts of it, can be costly; this is sometimes avoided by using "pointers" to data instead. Computations include simple operations such as incrementing the value of a variable data element. More complex computations may involve many operations and data elements together.

Quality and reliability

Main articles: Software quality and Software testing

Software quality is very important, especially for commercial and system software. If software is faulty, it can delete a person's work, crash the computer and do other unexpected things. Faults and errors are called "bugs" which are often discovered during alpha and beta testing. Software is often also a victim to what is known as software aging, the progressive performance degradation resulting from a combination of unseen bugs.

Many bugs are discovered and fixed through software testing. However, software testing rarely—if ever—eliminates every bug; some programmers say that "every program has at least one more bug" (Lubarsky's Law). In the waterfall method of software development, separate testing teams are typically employed, but in newer approaches, collectively termed agile software development, developers often do all their own testing, and demonstrate the software to users/clients regularly to obtain feedback. Software can be tested through unit testing, regression testing and other methods, which are done manually, or most commonly, automatically, since the amount of code to be tested can be large. Programs containing command software enable hardware engineering and system operations to function much easier together.

License

Main article: Software license

The software's license gives the user the right to use the software in the licensed environment, and in the case of free software licenses, also grants other rights such as the right to make copies.

Proprietary software can be divided into two types:

• freeware, which includes the category of "free trial" software or "freemium" software (in the past, the term shareware was often used for free trial/freemium software). As the name suggests, freeware can be used for free, although in the case of free trials or freemium software, this is sometimes only true for a limited period of time or with limited functionality.
• software available for a fee, which can only be legally used on purchase of a license.

Open-source software comes with a free software license, granting the recipient the rights to modify and redistribute the software.

Patents

Main articles: Software patent and Software patent debate

Software patents, like other types of patents, are theoretically supposed to give an inventor an exclusive, time-limited license for a detailed idea (e.g. an algorithm) on how to implement a piece of software, or a component of a piece of software. Ideas for useful things that software could do, and user requirements, are not supposed to be patentable, and concrete implementations (i.e. the actual software packages implementing the patent) are not supposed to be patentable either—the latter are already covered by copyright, generally automatically. So software patents are supposed to cover the middle area, between requirements and concrete implementation. In some countries, a requirement for the claimed invention to have an effect on the physical world may also be part of the requirements for a software patent to be held valid—although since all useful software has effects on the physical world, this requirement may be open to debate. Meanwhile, American copyright law was applied to various aspects of the writing of the software code.

Software patents are controversial in the software industry with many people holding different views about them. One of the sources of controversy is that the aforementioned split between initial ideas and patent does not seem to be honored in practice by patent lawyers—for example the patent for aspect-oriented programming (AOP), which purported to claim rights over any programming tool implementing the idea of AOP, howsoever implemented. Another source of controversy is the effect on innovation, with many distinguished experts and companies arguing that software is such a fast-moving field that software patents merely create vast additional litigation costs and risks, and actually retard innovation. In the case of debates about software patents outside the United States, the argument has been made that large American corporations and patent lawyers are likely to be the primary beneficiaries of allowing or continue to allow software patents.

Design and implementation

Main articles: Software development, Computer programming, and Software engineering

Design and implementation of software vary depending on the complexity of the software. For instance, the design and creation of Microsoft Word took much more time than designing and developing Microsoft Notepad because the former has much more basic functionality.

Software is usually developed in integrated development environments (IDE) like Eclipse, IntelliJ and Microsoft Visual Studio that can simplify the process and compile the software. As noted in a different section, software is usually created on top of existing software and the application programming interface (API) that the underlying software provides like GTK+, JavaBeans or Swing Libraries (APIs) can be categorized by their purpose. For instance, the Spring Framework is used for implementing enterprise applications, the Windows Forms library is used for designing graphical user interface (GUI) applications like Microsoft Word, and Windows Communication Foundation is used for designing web services. When a program is designed, it relies upon the API. For instance, a Microsoft Windows desktop application might call API functions in the .NET Windows Forms library like Form1.Close() and Form1.Show() to close or open the application. Without these APIs, programmers need to write these functionalities entirely themselves. Companies like Oracle and Microsoft provide their own APIs so that many applications are written using their software libraries that usually have numerous APIs in them.

Data structures such as hash tables, arrays, and binary trees, and algorithms such as quicksort, can be useful for creating software.

Computer software has special economic characteristics that make its design, creation, and distribution different from most other economic goods.

A person who creates software is called a programmer, software engineer or software developer, terms that all have a similar meaning. More informal terms for programmer also exist such as "coder" and "hacker" – although use of the latter word may cause confusion, because it is more often used to mean someone who illegally breaks into computer systems.

Software release life cycle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Software_release_life_cycle

The software release life cycle is the process of developing, testing, and distributing a software product. It typically consists of several stages, such as pre-alpha, alpha, beta, and release candidate, before the final version, or "gold," is released to the public.

An example of a basic software release life cycle

Pre-alpha refers to the early stages of development, when the software is still being designed and built. Alpha testing is the first phase of formal testing, during which the software is tested internally using white-box techniques. Beta testing is the next phase, in which the software is tested by a larger group of users, typically outside of the organization that developed it. The beta phase is focused on reducing impacts on users and may include usability testing.

After beta testing, the software may go through one or more release candidate phases, in which it is refined and tested further, before the final version is released.

Some software, particularly in the internet and technology industries, is released in a perpetual beta state, meaning that it is continuously being updated and improved, and is never considered to be a fully completed product. This approach allows for a more agile development process and enables the software to be released and used by users earlier in the development cycle.

Stages of development

Pre-alpha

Pre-alpha refers to all activities performed during the software project before formal testing. These activities can include requirements analysis, software design, software development, and unit testing. In typical open source development, there are several types of pre-alpha versions. Milestone versions include specific sets of functions and are released as soon as the feature is complete.

Alpha

The alpha phase of the release life cycle is the first phase of software testing (alpha is the first letter of the Greek alphabet, used as the number 1). In this phase, developers generally test the software using white-box techniques. Additional validation is then performed using black-box or gray-box techniques, by another testing team. Moving to black-box testing inside the organization is known as alpha release.

Alpha software is not thoroughly tested by the developer before it is released to customers. Alpha software may contain serious errors, and any resulting instability could cause crashes or data loss. Alpha software may not contain all of the features that are planned for the final version. In general, external availability of alpha software is uncommon in proprietary software, while open source software often has publicly available alpha versions. The alpha phase usually ends with a feature freeze, indicating that no more features will be added to the software. At this time, the software is said to be feature complete. A beta test is carried out following acceptance testing at the supplier's site (alpha test) and immediately before the general release of the software as a product.

In general, an alpha version or release of a software package intends to do something particular, and mostly does so, yet is not guaranteed to do so fully.

Feature complete

A feature complete (FC) version of a piece of software has all of its planned or primary features implemented but is not yet final due to bugs, performance or stability issues. This occurs at the end of alpha testing of development.

Usually, a feature-complete software still has to undergo beta testing and bug fixing, as well as performance or stability enhancement before it can go to release candidate, and finally gold status.

Beta

"Beta test" redirects here. For the 2016 film, see Beta Test (film). For the 2021 film, see The Beta Test.

See also: Software testing § Beta testing

Beta, named after the second letter of the Greek alphabet, is the software development phase following alpha. Software in the beta stage is also known as beta ware. A beta phase generally begins when the software is feature complete but likely to contain several known or unknown bugs. Software in the beta phase will generally have many more bugs in it than completed software and speed or performance issues, and may still cause crashes or data loss. The focus of beta testing is reducing impacts on users, often incorporating usability testing. The process of delivering a beta version to the users is called beta release and is typically the first time that the software is available outside of the organization that developed it. Software beta releases can be either open or closed, depending on whether they are openly available or only available to a limited audience. Beta version software is often useful for demonstrations and previews within an organization and to prospective customers. Some developers refer to this stage as a preview, preview release, prototype, technical preview or technology preview (TP), or early access.

Beta testers are people who actively report issues with beta software. They are usually customers or representatives of prospective customers of the organization that develops the software. Beta testers tend to volunteer their services free of charge but often receive versions of the product they test, discounts on the release version, or other incentives.

Perpetual beta

Main article: Perpetual beta

Some software is kept in so-called perpetual beta, where new features are continually added to the software without establishing a final "stable" release. As the Internet has facilitated the rapid and inexpensive distribution of software, companies have begun to take a looser approach to the use of the word beta.

Open and closed beta

Developers may release either a closed beta, or an open beta; closed beta versions are released to a restricted group of individuals for a user test by invitation, while open beta testers are from a larger group, or anyone interested. Private beta could be suitable for the software that is capable of delivering value but is not ready to be used by everyone either due to scaling issues, lack of documentation or still missing vital features. The testers report any bugs that they find, and sometimes suggest additional features they think should be available in the final version.

Open betas serve the dual purpose of demonstrating a product to potential consumers, and testing among a wide user base is likely to bring to light obscure errors that a much smaller testing team might not find.

Release candidate

A release candidate (RC), also known as gamma testing or "going silver", is a beta version with the potential to be a stable product, which is ready to release unless significant bugs emerge. In this stage of product stabilization, all product features have been designed, coded, and tested through one or more beta cycles with no known showstopper-class bugs. A release is called code complete when the development team agrees that no entirely new source code will be added to this release. There could still be source code changes to fix defects, changes to documentation and data files, and peripheral code for test cases or utilities. Beta testers, if privately selected, will often be credited for using the release candidate as though it were a finished product. Beta testing is conducted in a client's or customer's location and to test the software from a user's perspective.

Stable release

Also called production release, the stable release is the last release candidate (RC) which has passed all stages of verification and tests. The remaining bugs are considered acceptable. This release goes to production.

Some software products (e.g. Linux distributions) also have long term support (LTS) releases which are based on full releases that have already been tried and tested and receive only security updates. This allows developers to allocate more time toward product development instead of updating code or finding and fixing newly introduced bugs due to outdated assumptions about the used system, language, or underlying libraries.

Release

Once released, the software is generally known as a "stable release". The formal term often depends on the method of release: physical media, online release, or a web application.

Release to manufacturing (RTM)

The term release to manufacturing (RTM), also known as "going gold", is a term used when a software product is ready to be delivered. This build may be digitally signed, allowing the end user to verify the integrity and authenticity of the software purchase. A copy of the RTM build known as the "gold master" or GM is sent for mass duplication or disc replication if applicable. The terminology is taken from the audio record-making industry, specifically the process of mastering. RTM precedes general availability (GA) when the product is released to the public. A golden master build (GM) is typically the final build of a piece of software in the beta stages for developers. Typically, for iOS, it is the final build before a major release, however, there have been a few exceptions.

RTM is typically used in certain retail mass-production software contexts—as opposed to a specialized software production or project in a commercial or government production and distribution—where the software is sold as part of a bundle in a related computer hardware sale and typically where the software and related hardware is ultimately to be available and sold on mass/public basis at retail stores to indicate that the software has met a defined quality level and is ready for mass retail distribution. RTM could also mean in other contexts that the software has been delivered or released to a client or customer for installation or distribution to the related hardware end user computers or machines. The term does not define the delivery mechanism or volume; it only states that the quality is sufficient for mass distribution. The deliverable from the engineering organization is frequently in the form of a golden master media used for duplication or to produce the image for the web.

General availability (GA)

Milestones in a product life cycle: general availability (GA), end of life announcement (EOLA), last order date (LOD), and end-of-life (EOL)

General availability (GA) is the marketing stage at which all necessary commercialization activities have been completed and a software product is available for purchase, depending, however, on language, region, and electronic vs. media availability. Commercialization activities could include security and compliance tests, as well as localization and worldwide availability. The time between RTM and GA can take from days to months before a generally available release can be declared, due to the time needed to complete all commercialization activities required by GA. At this stage, the software has "gone live".

Release to the Web (RTW)

Release to the Web (RTW) or Web release is a means of software delivery that utilizes the Internet for distribution. No physical media are produced in this type of release mechanism by the manufacturer. Web releases have become more common as Internet usage grew.

Support

During its supported lifetime, the software is sometimes subjected to service releases, patches or service packs, sometimes also called "interim releases" or "maintenance releases" (MR). For example, Microsoft released three major service packs for the 32-bit editions of Windows XP and two service packs for the 64-bit editions. Such service releases contain a collection of updates, fixes, and enhancements, delivered in the form of a single installable package. They may also implement new features. Some software is released with the expectation of regular support. Classes of software that generally involve protracted support as the norm include anti-virus suites and massively multiplayer online games. Continuing with this Windows XP example, Microsoft did offer paid updates for five more years after the end of extended support. This means that support ended on April 8, 2019.

End-of-life

See also: End-of-life product and Abandonware

When software is no longer sold or supported, the product is said to have reached end-of-life, to be discontinued, retired, deprecated, abandoned, or obsolete, but user loyalty may continue its existence for some time, even long after its platform is obsolete—e.g., the Common Desktop Environment and Sinclair ZX Spectrum.

After the end-of-life date, the developer will usually not implement any new features, fix existing defects, bugs, or vulnerabilities (whether known before that date or not), or provide any support for the product. If the developer wishes, they may release the source code, so the platform will live again, and be maintained by volunteers, and if not, it may be reverse-engineered later when it becomes abandonware.

History

Usage of the "alpha/beta" test terminology originated at IBM. Similar terminologies for IBM's software development were used by people involved with IBM from at least the 1950s (and probably earlier). "A" test was the verification of a new product before the public announcement. The "B" test was the verification before releasing the product to be manufactured. The "C" test was the final test before the general availability of the product. As software became a significant part of IBM's offerings, the alpha test terminology was used to denote the pre-announcement test and the beta test was used to show product readiness for general availability. Martin Belsky, a manager on some of IBM's earlier software projects claimed to have invented the terminology. IBM dropped the alpha/beta terminology during the 1960s, but by then it had received fairly wide notice. The usage of "beta test" to refer to testing done by customers was not done in IBM. Rather, IBM used the term "field test".

Major public beta's developed afterward, with early customers having purchased a "pioneer edition" of the WordVision word processor for the IBM PC for $49.95. In 1984, Stephen Manes wrote that "in a brilliant marketing coup, Bruce and James Program Publishers managed to get people to pay for the privilege of testing the product." In September 2000 a boxed version of Apple's Mac OS X Public Beta operating system was released. Microsoft's release of community technology previews (CTPs) for Windows Vista, between September 2005 and May 2006. and from 2009 to 2011, Minecraft was in public beta.

In February 2005, ZDNet published an article about the phenomenon of a beta version often staying for years and being used as if it were at the production level. It noted that Gmail and Google News, for example, had been in beta for a long time although widely used; Google News left beta in January 2006, followed by Google Apps (now named Google Workspace), including Gmail, in July 2009. Since the introduction of Windows 8, Microsoft has called pre-release software a preview rather than beta. All pre-release builds released through the Windows Insider Program launched in 2014 are termed "Insider Preview builds". "Beta" may also indicate something more like a release candidate, or as a form of time-limited demo, or marketing technique.

Degenerate energy levels

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Degenerate_energy_levels

In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

Degeneracy plays a fundamental role in quantum statistical mechanics. For an N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level all have an equal probability of being filled. The number of such states gives the degeneracy of a particular energy level.

Degenerate states in a quantum system

Mathematics

The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If A is a N × N matrix, X a non-zero vector, and λ is a scalar, such that ${\displaystyle AX=\lambda X}$, then the scalar λ is said to be an eigenvalue of A and the vector X is said to be the eigenvector corresponding to λ. Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue λ form a subspace of Cⁿ, which is called the eigenspace of λ. An eigenvalue λ which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e., ${\displaystyle A{X}_1=\lambda X_1}$ and ${\displaystyle A{X}_2=\lambda X_2}$, where ${\displaystyle X_1}$ and ${\displaystyle X_2}$ are linearly independent eigenvectors. The dimension of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional.

The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue. This clearly follows from the fact that the eigenspace of the energy value eigenvalue λ is a subspace (being the kernel of the Hamiltonian minus λ times the identity), hence is closed under linear combinations.

Proof of the above theorem.

If ${\displaystyle \hat{H}}$ represents the Hamiltonian operator and ${\displaystyle |{\psi }_1⟩}$ and ${\displaystyle |{\psi }_2⟩}$ are two eigenstates corresponding to the same eigenvalue E, then

$${\displaystyle \hat{H}|{\psi }_1⟩=E|{\psi }_1⟩}$$

$${\displaystyle \hat{H}|{\psi }_2⟩=E|{\psi }_2⟩}$$

Let ${\displaystyle |\psi ⟩=c_1|{\psi }_1⟩+c_2|{\psi }_2⟩}$, where ${\displaystyle c_1}$ and ${\displaystyle c_2}$ are complex(in general) constants, be any linear combination of ${\displaystyle |{\psi }_1⟩}$ and ${\displaystyle |{\psi }_2⟩}$. Then,

$${\displaystyle \begin{array}{rl}\hat{H}|\psi ⟩& =\hat{H}(c_1|{\psi }_1⟩+c_2|{\psi }_2⟩)\\ & =c_1\hat{H}|{\psi }_1⟩+c_2\hat{H}|{\psi }_2⟩\\ & =E(c_1|{\psi }_1⟩+c_2|{\psi }_2⟩)\\ & =E|\psi ⟩\end{array}}$$

which shows that ${\displaystyle |\psi ⟩}$ is an eigenstate of ${\displaystyle \hat{H}}$ with the same eigenvalue E.

Effect of degeneracy on the measurement of energy

In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. However, if the Hamiltonian ${\displaystyle \hat{H}}$ has a degenerate eigenvalue ${\displaystyle E_n}$ of degree g_n, the eigenstates associated with it form a vector subspace of dimension g_n. In such a case, several final states can be possibly associated with the same result ${\displaystyle E_n}$, all of which are linear combinations of the g_n orthonormal eigenvectors ${\displaystyle |E_{n,i}⟩}$.

In this case, the probability that the energy value measured for a system in the state ${\displaystyle |\psi ⟩}$ will yield the value ${\displaystyle E_n}$ is given by the sum of the probabilities of finding the system in each of the states in this basis, i.e.

${\displaystyle P(E_n)=\sum _{i=1}^{g_n}|⟨E_{n,i}|\psi ⟩{|}^2}$

Degeneracy in different dimensions

This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual understanding of more complex systems.

Degeneracy in one dimension

In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function ${\displaystyle |\psi ⟩}$ moving in a one-dimensional potential ${\displaystyle V(x)}$, the time-independent Schrödinger equation can be written as

${\displaystyle -\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+V\psi =E\psi }$

Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy ${\displaystyle E}$ at most, so that the degree of degeneracy never exceeds two. It can be proven that in one dimension, there are no degenerate bound states for normalizable wave functions. A sufficient condition on a piecewise continuous potential ${\displaystyle V}$ and the energy ${\displaystyle E}$ is the existence of two real numbers ${\displaystyle M,x_0}$ with ${\displaystyle M\ne 0}$ such that ${\displaystyle \mathrm{\forall }x>x_0}$ we have ${\displaystyle V(x)-E\ge M^2}$. In particular, ${\displaystyle V}$ is bounded below in this criterion.

Degeneracy in two-dimensional quantum systems

Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made of monoatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems.

Particle in a rectangular plane

Consider a free particle in a plane of dimensions ${\displaystyle L_x}$ and ${\displaystyle L_y}$ in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with wave function ${\displaystyle |\psi ⟩}$ can be written as

${\displaystyle -\frac{{\hslash }^2}{2m}\left(\frac{{\mathrm{\partial }}^2\psi }{{\mathrm{\partial }x}^2}+\frac{{\mathrm{\partial }}^2\psi }{{\mathrm{\partial }y}^2}\right)=E\psi }$

The permitted energy values are

${\displaystyle E_{n_x,n_y}=\frac{{\pi }^2{\hslash }^2}{2m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}\right)}$

The normalized wave function is

${\displaystyle {\psi }_{n_x,n_y}(x,y)=\frac{2}{\sqrt{L_x{L}_y}}\sin \left(\frac{n_x\pi x}{L_x}\right)\sin \left(\frac{n_y\pi y}{L_y}\right)}$

where ${\displaystyle n_x,n_y=1,2,\mathrm{3...}}$

So, quantum numbers ${\displaystyle n_x}$ and ${\displaystyle n_y}$ are required to describe the energy eigenvalues and the lowest energy of the system is given by

${\displaystyle E_{1,1}={\pi }^2\frac{{\hslash }^2}{2m}\left(\frac{1}{L_x^2}+\frac{1}{L_y^2}\right)}$

For some commensurate ratios of the two lengths ${\displaystyle L_x}$ and ${\displaystyle L_y}$, certain pairs of states are degenerate. If ${\displaystyle L_x/L_y=p/q}$, where p and q are integers, the states ${\displaystyle (n_x,n_y)}$ and ${\displaystyle (p{n}_y/q,q{n}_x/p)}$ have the same energy and so are degenerate to each other.

Particle in a square box

In this case, the dimensions of the box ${\displaystyle L_x=L_y=L}$ and the energy eigenvalues are given by

${\displaystyle E_{n_x,n_y}=\frac{{\pi }^2{\hslash }^2}{2m{L}^2}(n_x^2+n_y^2)}$

Since ${\displaystyle n_x}$ and ${\displaystyle n_y}$ can be interchanged without changing the energy, each energy level has a degeneracy of at least two when ${\displaystyle n_x}$ and ${\displaystyle n_y}$ are different. Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For example, the three states (n_x = 7, n_y = 1), (n_x = 1, n_y = 7) and (n_x = n_y = 5) all have ${\displaystyle E=50\frac{{\pi }^2{\hslash }^2}{2m{L}^2}}$ and constitute a degenerate set.

Degrees of degeneracy of different energy levels for a particle in a square box:

${\displaystyle n_x}$	${\displaystyle n_y}$	${\displaystyle E\left(\frac{{\hslash }^2{\pi }^2}{2m{L}^2}\right)}$	Degeneracy
1	1	2	1
2 1	1 2	5 5	2
2	2	8	1
3 1	1 3	10 10	2
3 2	2 3	13 13	2
4 1	1 4	17 17	2
3	3	18	1
...	...	...	...
7 5 1	1 5 7	50 50 50	3
...	...	...	...
8 7 4 1	1 4 7 8	65 65 65 65	4
...	...	...	...
9 7 6 2	2 6 7 9	85 85 85 85	4
...	...	...	...
11 10 5 2	2 5 10 11	125 125 125 125	4
...	...	...	...
14 10 2	2 10 14	200 200 200	3
...	...	...	...
17 13 7	7 13 17	338 338 338	3

Particle in a cubic box

In this case, the dimensions of the box ${\displaystyle L_x=L_y=L_z=L}$ and the energy eigenvalues depend on three quantum numbers.

${\displaystyle E_{n_x,n_y,n_z}=\frac{{\pi }^2{\hslash }^2}{2m{L}^2}(n_x^2+n_y^2+n_z^2)}$

Since ${\displaystyle n_x}$, ${\displaystyle n_y}$ and ${\displaystyle n_z}$ can be interchanged without changing the energy, each energy level has a degeneracy of at least three when the three quantum numbers are not all equal.

Finding a unique eigenbasis in case of degeneracy

If two operators ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$ commute, i.e. ${\displaystyle [\hat{A},\hat{B}]=0}$, then for every eigenvector ${\displaystyle |\psi ⟩}$ of ${\displaystyle \hat{A}}$, ${\displaystyle \hat{B}|\psi ⟩}$ is also an eigenvector of ${\displaystyle \hat{A}}$ with the same eigenvalue. However, if this eigenvalue, say ${\displaystyle \lambda }$, is degenerate, it can be said that ${\displaystyle \hat{B}|\psi ⟩}$ belongs to the eigenspace ${\displaystyle E_{\lambda }}$ of ${\displaystyle \hat{A}}$, which is said to be globally invariant under the action of ${\displaystyle \hat{B}}$.

For two commuting observables A and B, one can construct an orthonormal basis of the state space with eigenvectors common to the two operators. However, ${\displaystyle \lambda }$ is a degenerate eigenvalue of ${\displaystyle \hat{A}}$, then it is an eigensubspace of ${\displaystyle \hat{A}}$ that is invariant under the action of ${\displaystyle \hat{B}}$, so the representation of ${\displaystyle \hat{B}}$ in the eigenbasis of ${\displaystyle \hat{A}}$ is not a diagonal but a block diagonal matrix, i.e. the degenerate eigenvectors of ${\displaystyle \hat{A}}$ are not, in general, eigenvectors of ${\displaystyle \hat{B}}$. However, it is always possible to choose, in every degenerate eigensubspace of ${\displaystyle \hat{A}}$, a basis of eigenvectors common to ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$.

Choosing a complete set of commuting observables

If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. On the other hand, if one or several eigenvalues of ${\displaystyle \hat{A}}$ are degenerate, specifying an eigenvalue is not sufficient to characterize a basis vector. If, by choosing an observable ${\displaystyle \hat{B}}$, which commutes with ${\displaystyle \hat{A}}$, it is possible to construct an orthonormal basis of eigenvectors common to ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$, which is unique, for each of the possible pairs of eigenvalues {a,b}, then ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$ are said to form a complete set of commuting observables. However, if a unique set of eigenvectors can still not be specified, for at least one of the pairs of eigenvalues, a third observable ${\displaystyle \hat{C}}$, which commutes with both ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$ can be found such that the three form a complete set of commuting observables.

It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator that commutes with the Hamiltonian. These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system.

Degenerate energy eigenstates and the parity operator

The parity operator is defined by its action in the ${\displaystyle |r⟩}$ representation of changing r to −r, i.e.

${\displaystyle ⟨r|P|\psi ⟩=\psi (-r)}$

The eigenvalues of P can be shown to be limited to ${\displaystyle \pm 1}$, which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while that with eigenvalue −1 is said to be odd.

Now, an even operator ${\displaystyle \hat{A}}$ is one that satisfies,

${\displaystyle \tilde{A}=P\hat{A}P}$

${\displaystyle [P,\hat{A}]=0}$

while an odd operator ${\displaystyle \hat{B}}$ is one that satisfies

${\displaystyle P\hat{B}+\hat{B}P=0}$

Since the square of the momentum operator ${\displaystyle {\hat{p}}^2}$ is even, if the potential V(r) is even, the Hamiltonian ${\displaystyle \hat{H}}$ is said to be an even operator. In that case, if each of its eigenvalues are non-degenerate, each eigenvector is necessarily an eigenstate of P, and therefore it is possible to look for the eigenstates of ${\displaystyle \hat{H}}$ among even and odd states. However, if one of the energy eigenstates has no definite parity, it can be asserted that the corresponding eigenvalue is degenerate, and ${\displaystyle P|\psi ⟩}$ is an eigenvector of ${\displaystyle \hat{H}}$ with the same eigenvalue as ${\displaystyle |\psi ⟩}$.

Degeneracy and symmetry

The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find the energy levels and degeneracies without solving the Schrödinger equation, hence reducing effort.

Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that ${\displaystyle H^{\prime }=SH{S}^{-1}=SH{S}^{†}}$, since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, we have

${\displaystyle SH{S}^{†}=H}$

${\displaystyle SH{S}^{-1}=H}$

${\displaystyle SH=HS}$

${\displaystyle [S,H]=0}$

Now, if ${\displaystyle |\alpha ⟩}$ is an energy eigenstate,

${\displaystyle H|\alpha ⟩=E|\alpha ⟩}$

where E is the corresponding energy eigenvalue.

${\displaystyle HS|\alpha ⟩=SH|\alpha ⟩=SE|\alpha ⟩=ES|\alpha ⟩}$

which means that ${\displaystyle S|\alpha ⟩}$ is also an energy eigenstate with the same eigenvalue E. If the two states ${\displaystyle |\alpha ⟩}$ and ${\displaystyle S|\alpha ⟩}$ are linearly independent (i.e. physically distinct), they are therefore degenerate.

In cases where S is characterized by a continuous parameter ${\displaystyle ϵ}$, all states of the form ${\displaystyle S(ϵ)|\alpha ⟩}$ have the same energy eigenvalue.

Symmetry group of the Hamiltonian

The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.

Types of degeneracy

Degeneracies in a quantum system can be systematic or accidental in nature.

Systematic or essential degeneracy

This is also called a geometrical or normal degeneracy and arises due to the presence of some kind of symmetry in the system under consideration, i.e. the invariance of the Hamiltonian under a certain operation, as described above. The representation obtained from a normal degeneracy is irreducible and the corresponding eigenfunctions form a basis for this representation.

Accidental degeneracy

It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in the system.^[4] It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete. These degeneracies are connected to the existence of bound orbits in classical Physics.

Examples: Coulomb and Harmonic Oscillator potentials

For a particle in a central 1/r potential, the Laplace–Runge–Lenz vector is a conserved quantity resulting from an accidental degeneracy, in addition to the conservation of angular momentum due to rotational invariance.

For a particle moving on a cone under the influence of 1/r and r² potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of an equivalent of the Runge-Lenz vector, in addition to one component of the angular momentum vector. These quantities generate SU(2) symmetry for both potentials.

Example: Particle in a constant magnetic field

A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry. The symmetry multiplets in this case are the Landau levels which are infinitely degenerate.

Examples

The hydrogen atom

Main article: Hydrogen Atom

In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total orbital angular momentum ${\displaystyle \widehat{L^2}}$, its component along the z-direction, ${\displaystyle \widehat{L_z}}$, total spin angular momentum ${\displaystyle \widehat{S^2}}$ and its z-component ${\displaystyle \widehat{S_z}}$. The quantum numbers corresponding to these operators are ${\displaystyle l}$, ${\displaystyle m_l}$, ${\displaystyle s}$ (always 1/2 for an electron) and ${\displaystyle m_s}$ respectively.

The energy levels in the hydrogen atom depend only on the principal quantum number n. For a given n, all the states corresponding to ${\displaystyle l=0,\dots ,n-1}$ have the same energy and are degenerate. Similarly for given values of n and l, the ${\displaystyle (2l+1)}$, states with ${\displaystyle m_l=-l,\dots ,l}$ are degenerate. The degree of degeneracy of the energy level E_n is therefore :${\displaystyle \sum _{l=0}^{n-1}(2l+1)=n^2}$, which is doubled if the spin degeneracy is included.

The degeneracy with respect to ${\displaystyle m_l}$ is an essential degeneracy which is present for any central potential, and arises from the absence of a preferred spatial direction. The degeneracy with respect to ${\displaystyle l}$ is often described as an accidental degeneracy, but it can be explained in terms of special symmetries of the Schrödinger equation which are only valid for the hydrogen atom in which the potential energy is given by Coulomb's law.

Isotropic three-dimensional harmonic oscillator

It is a spinless particle of mass m moving in three-dimensional space, subject to a central force whose absolute value is proportional to the distance of the particle from the centre of force.

${\displaystyle F=-kr}$

It is said to be isotropic since the potential ${\displaystyle V(r)}$ acting on it is rotationally invariant, i.e. :${\displaystyle V(r)=1/2\left(m{\omega }^2{r}^2\right)}$

where ${\displaystyle \omega }$ is the angular frequency given by $\sqrt{k/m}$.

Since the state space of such a particle is the tensor product of the state spaces associated with the individual one-dimensional wave functions, the time-independent Schrödinger equation for such a system is given by-

${\displaystyle -\frac{{\hslash }^2}{2m}\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{z}^2}\right)+\frac{1}{2}m{\omega }^2(x^2+y^2+z^2)\psi =E\psi }$

So, the energy eigenvalues are ${\displaystyle E_{n_x,n_y,n_z}=(n_x+n_y+n_z+3/2)\hslash \omega }$

or, ${\displaystyle E_n=(n+3/2)\hslash \omega }$

where n is a non-negative integer. So, the energy levels are degenerate and the degree of degeneracy is equal to the number of different sets ${\displaystyle \{n_x,n_y,n_z\}}$ satisfying

${\displaystyle n_x+n_y+n_z=n}$

The degeneracy of the ${\displaystyle n}$-th state can be found by considering the distribution of ${\displaystyle n}$ quanta across ${\displaystyle n_x}$, ${\displaystyle n_y}$ and ${\displaystyle n_z}$. Having 0 in ${\displaystyle n_x}$ gives ${\displaystyle n+1}$ possibilities for distribution across ${\displaystyle n_y}$ and ${\displaystyle n_z}$. Having 1 quanta in ${\displaystyle n_x}$ gives ${\displaystyle n}$ possibilities across ${\displaystyle n_y}$ and ${\displaystyle n_z}$ and so on. This leads to the general result of ${\displaystyle n-n_x+1}$ and summing over all ${\displaystyle n}$ leads to the degeneracy of the ${\displaystyle n}$-th state,

${\displaystyle \sum _{n_x=0}^n(n-n_x+1)=\frac{(n+1)(n+2)}{2}}$

As shown, only the ground state where ${\displaystyle n=0}$ is non-degenerate (ie, has a degeneracy of ${\displaystyle 1}$).

Removing degeneracy

The degeneracy in a quantum mechanical system may be removed if the underlying symmetry is broken by an external perturbation. This causes splitting in the degenerate energy levels. This is essentially a splitting of the original irreducible representations into lower-dimensional such representations of the perturbed system.

Mathematically, the splitting due to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation scheme that can be applied to find the solution to the eigenvalue equation for the Hamiltonian H of a quantum system with an applied perturbation, given the solution for the Hamiltonian H₀ for the unperturbed system. It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the perturbed system near them. The correct basis to choose is one that diagonalizes the perturbation Hamiltonian within the degenerate subspace.

Physical examples of removal of degeneracy by a perturbation

Some important examples of physical situations where degenerate energy levels of a quantum system are split by the application of an external perturbation are given below.

Symmetry breaking in two-level systems

See also: Avoided crossing § In two state systems

A two-level system essentially refers to a physical system having two states whose energies are close together and very different from those of the other states of the system. All calculations for such a system are performed on a two-dimensional subspace of the state space.

If the ground state of a physical system is two-fold degenerate, any coupling between the two corresponding states lowers the energy of the ground state of the system, and makes it more stable.

If ${\displaystyle E_1}$ and ${\displaystyle E_2}$ are the energy levels of the system, such that ${\displaystyle E_1=E_2=E}$, and the perturbation ${\displaystyle W}$ is represented in the two-dimensional subspace as the following 2×2 matrix

${\displaystyle \mathbf{W}=\left[\begin{array}{cc}0& W_{12}\\ W_{12}^{\ast }& 0\end{array}\right].}$

then the perturbed energies are

${\displaystyle E_{+}=E+|W_{12}|}$

${\displaystyle E_{-}=E-|W_{12}|}$

Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent property of the system include:

• Benzene, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
• Ammonia molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.
• H⁺
  ₂ molecule, in which the electron may be localized around either of the two nuclei.

Fine-structure splitting

Main article: Fine structure

The corrections to the Coulomb interaction between the electron and the proton in a Hydrogen atom due to relativistic motion and spin–orbit coupling result in breaking the degeneracy in energy levels for different values of l corresponding to a single principal quantum number n.

The perturbation Hamiltonian due to relativistic correction is given by

${\displaystyle H_r=-p^4/8m^3{c}^2}$

where ${\displaystyle p}$ is the momentum operator and ${\displaystyle m}$ is the mass of the electron. The first-order relativistic energy correction in the ${\displaystyle |nlm⟩}$ basis is given by

${\displaystyle E_r=(-1/8m^3{c}^2)⟨nlm|p^4|nlm⟩}$

Now ${\displaystyle p^4=4m^2(H^0+e^2/r{)}^2}$

${\displaystyle \begin{array}{rl}{E}_r& =(-1/2m{c}^2)[E_n^2+2E_n{e}^2⟨1/r⟩+e^4⟨1/r^2⟩]\\ & =(-1/2)m{c}^2{\alpha }^4[-3/(4n^4)+1/n^3(l+1/2)]\end{array}}$

where ${\displaystyle \alpha }$ is the fine structure constant.

The spin–orbit interaction refers to the interaction between the intrinsic magnetic moment of the electron with the magnetic field experienced by it due to the relative motion with the proton. The interaction Hamiltonian is

${\displaystyle H_{so}=-(e/mc)\overrightarrow{m}\cdot \overrightarrow{L}/r^3=[(e^2/(m^2{c}^2{r}^3))\overrightarrow{S}\cdot \overrightarrow{L}]}$

which may be written as

${\displaystyle H_{so}=(e^2/(4m^2{c}^2{r}^3))[{\overrightarrow{J}}^2-{\overrightarrow{L}}^2-{\overrightarrow{S}}^2]}$

The first order energy correction in the ${\displaystyle |j,m,l,1/2⟩}$ basis where the perturbation Hamiltonian is diagonal, is given by

${\displaystyle E_{so}=({\hslash }^2{e}^2)/(4m^2{c}^2)[j(j+1)-l(l+1)-3/4]/((a_0{)}^3{n}^3(l(l+1/2)(l+1))]}$

where ${\displaystyle a_0}$ is the Bohr radius. The total fine-structure energy shift is given by

${\displaystyle E_{fs}=-(m{c}^2{\alpha }^4/(2n^3))[1/(j+1/2)-3/4n]}$

for ${\displaystyle j=l\pm 1/2}$.

Zeeman effect

Main article: Zeeman effect

The splitting of the energy levels of an atom when placed in an external magnetic field because of the interaction of the magnetic moment ${\displaystyle \overrightarrow{m}}$ of the atom with the applied field is known as the Zeeman effect.

Taking into consideration the orbital and spin angular momenta, ${\displaystyle \overrightarrow{L}}$ and ${\displaystyle \overrightarrow{S}}$, respectively, of a single electron in the Hydrogen atom, the perturbation Hamiltonian is given by

${\displaystyle \hat{V}=-(\overrightarrow{m_l}+\overrightarrow{m_s})\cdot \overrightarrow{B}}$

where ${\displaystyle m_l=-e\overrightarrow{L}/2m}$ and ${\displaystyle m_s=-e\overrightarrow{S}/m}$. Thus,

${\displaystyle \hat{V}=e(\overrightarrow{L}+2\overrightarrow{S})\cdot \overrightarrow{B}/2m}$

Now, in case of the weak-field Zeeman effect, when the applied field is weak compared to the internal field, the spin–orbit coupling dominates and ${\displaystyle \overrightarrow{L}}$ and ${\displaystyle \overrightarrow{S}}$ are not separately conserved. The good quantum numbers are n, l, j and m_j, and in this basis, the first order energy correction can be shown to be given by

${\displaystyle E_z=-{\mu }_B{g}_{j}B{m}_j}$, where

${\displaystyle {\mu }_B=e\hslash /2m}$ is called the Bohr Magneton.Thus, depending on the value of ${\displaystyle m_j}$, each degenerate energy level splits into several levels.

Lifting of degeneracy by an external magnetic field

In case of the strong-field Zeeman effect, when the applied field is strong enough, so that the orbital and spin angular momenta decouple, the good quantum numbers are now n, l, m_l, and m_s. Here, L_z and S_z are conserved, so the perturbation Hamiltonian is given by-

${\displaystyle \hat{V}=eB(L_z+2S_z)/2m}$

assuming the magnetic field to be along the z-direction. So,

${\displaystyle \hat{V}=eB(m_l+2m_s)/2m}$

For each value of m_l, there are two possible values of m_s, ${\displaystyle \pm 1/2}$.

Stark effect

Main article: Stark effect

The splitting of the energy levels of an atom or molecule when subjected to an external electric field is known as the Stark effect.

For the hydrogen atom, the perturbation Hamiltonian is

${\displaystyle {\hat{H}}_s=-|e|Ez}$

if the electric field is chosen along the z-direction.

The energy corrections due to the applied field are given by the expectation value of ${\displaystyle {\hat{H}}_s}$ in the ${\displaystyle |nlm⟩}$ basis. It can be shown by the selection rules that ${\displaystyle ⟨nl{m}_l|z|n_1{l}_1{m}_{l1}⟩\ne 0}$ when ${\displaystyle l=l_1\pm 1}$ and ${\displaystyle m_l=m_{l1}}$.

The degeneracy is lifted only for certain states obeying the selection rules, in the first order. The first-order splitting in the energy levels for the degenerate states ${\displaystyle |2,0,0⟩}$ and ${\displaystyle |2,1,0⟩}$, both corresponding to n = 2, is given by ${\displaystyle \mathrm{\Delta }{E}_{2,1,m_l}=\pm |e|({\hslash }^2)/(m_e{e}^2)E}$.