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Ground state
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For articles related to the ground state, see Quantum vacuum (disambiguation).
Energy levels for an electron in an atom: ground state and excited states. After absorbing energy, an electron may jump from the ground state to a higher energy excited state.
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In the quantum field theory, the ground state is usually called the vacuum state or the vacuum.
If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system.
According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature.
Contents
[hide]
1 Ground state has no nodes in one-dimension
2 Examples
3 Notes
4 Bibliography
Ground state has no nodes in one-dimension[edit]
In one-dimension, the ground state of the Schrödinger equation can be proven to have no nodes.[1]
Consider the average energy of a state with a node at x = 0; i.e., ψ(0)=0. The average energy in this state would be:
⟨ ψ | H | ψ ⟩ = ∫ d x ( − ℏ 2 2 m ψ ∗ d 2 ψ d x 2 + V ( x ) | ψ ( x ) | 2 ) , {\displaystyle \left\langle \psi |H|\psi \right\rangle =\int dx\;\left(-{\frac {\hbar ^{2}}{2m}}\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}+V(x)|\psi (x)|^{2}\right)~,}
where V(x) is the potential.
Now, consider a small interval around x = 0 {\displaystyle x=0} ; i.e., x ∈ [ − ϵ , ϵ ] {\displaystyle x\in [-\epsilon ,\epsilon ]} . Take a new (deformed) wave function ψ' (x) to be defined as ψ ′ ( x ) = ψ ( x ) {\displaystyle \psi '(x)=\psi (x)} , for x < − ϵ {\displaystyle x<-\epsilon } ; and ψ ′ ( x ) = − ψ ( x ) {\displaystyle \psi '(x)=-\psi (x)} , for x > ϵ {\displaystyle x>\epsilon } ; and constant for x ∈ [ − ϵ , ϵ ] {\displaystyle x\in [-\epsilon ,\epsilon ]} . If ϵ {\displaystyle \epsilon } is small enough, this is always possible to do, so that ψ' (x) is continuous.
Assuming ψ ( x ) ≈ − c x {\displaystyle \psi (x)\approx -cx} around x = 0 {\displaystyle x=0} , one may write
ψ ′ ( x ) = N { | ψ ( x ) | | x | > ϵ c ϵ | x | ≤ ϵ {\displaystyle \psi '(x)=N\left\{{\begin{array}{ll}|\psi (x)|&|x|>\epsilon \\c\epsilon &|x|\leq \epsilon \end{array}}\right.}
where N = 1 1 + 4 3 | c | 2 ϵ 3 {\displaystyle N={\frac {1}{\sqrt {1+{\frac {4}{3}}|c|^{2}\epsilon ^{3}}}}} is the norm.
Note that the kinetic energy density | d ψ ′ d x | 2 < | d ψ d x | 2 {\displaystyle \left|{\frac {d\psi '}{dx}}\right|^{2}<\left|{\frac {d\psi }{dx}}\right|^{2}} everywhere because of the normalization. More significantly, the average kinetic energy is lowered by O ( ϵ ) {\displaystyle O(\epsilon )} by the deformation to ψ' .
Now, consider the potential energy. For definiteness, let us choose V ( x ) ≥ 0 {\displaystyle V(x)\geq 0} . Then it is clear that, outside the interval x ∈ [ − ϵ , ϵ ] {\displaystyle x\in [-\epsilon ,\epsilon ]} , the potential energy density is smaller for the ψ' because | ψ ′ | < | ψ | {\displaystyle |\psi '|<|\psi |} there.
On the other hand, in the interval x ∈ [ − ϵ , ϵ ] {\displaystyle x\in [-\epsilon ,\epsilon ]} we have
V avg ϵ ′ = ∫ − ϵ ϵ d x V ( x ) | ψ ′ | 2 = ϵ 2 | c | 2 1 + 4 3 | c | 2 ϵ 3 ∫ − ϵ ϵ V ( x ) ≃ 2 ϵ 3 | c | 2 V ( 0 ) + ⋯ , {\displaystyle {V_{\text{avg}}^{\epsilon }}'=\int _{-\epsilon }^{\epsilon }dx\;V(x)|\psi '|^{2}={\frac {\epsilon ^{2}|c|^{2}}{1+{\frac {4}{3}}|c|^{2}\epsilon ^{3}}}\int _{-\epsilon }^{\epsilon }V(x)\simeq 2\epsilon ^{3}|c|^{2}V(0)+\cdots ,}
which is holds to order ϵ 3 {\displaystyle \epsilon ^{3}} .
However, the contribution to the potential energy from this region for the state with a node, ψ, is
V avg ϵ = ∫ − ϵ ϵ d x V ( x ) | ψ | 2 = | c | 2 ∫ − ϵ ϵ d x x 2 V ( x ) ≃ 2 3 ϵ 3 | c | 2 V ( 0 ) + ⋯ , {\displaystyle V_{\text{avg}}^{\epsilon }=\int _{-\epsilon }^{\epsilon }dx\;V(x)|\psi |^{2}=|c|^{2}\int _{-\epsilon }^{\epsilon }dx\;x^{2}V(x)\simeq {\frac {2}{3}}\epsilon ^{3}|c|^{2}V(0)+\cdots ,}
lower, but still of the same lower order O ( ϵ 3 ) {\displaystyle O(\epsilon ^{3})} as for the deformed state ψ' , and subdominant to the lowering of the average kinetic energy. Therefore, the potential energy is unchanged up to order ϵ 2 {\displaystyle \epsilon ^{2}} , if we deform the state with a node ψ {\displaystyle \psi } into a state without a node ψ' , and the change can be ignored.
We can therefore remove all nodes and reduce the energy by O ( ϵ ) {\displaystyle O(\epsilon )} , which implies that ψ' cannot be the ground state: the ground-state wave function cannot have a node. This completes the proof. (The average energy may then be further lowered by eliminating undulations, to the variational absolute minimum.)
Examples[edit]
Initial wave functions for the first four states of a one-dimensional particle in a box
The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The energy of the particle is given by h 2 n 2 8 m L 2 {\displaystyle {\frac {h^{2}n^{2}}{8mL^{2}}}} , where h is the Planck constant, m is the mass of the particle, n is the energy state (n = 1 corresponds to the ground-state energy), and L is the width of the well.
The wave function of the ground state of a hydrogen atom is a spherically-symmetric distribution centred on the nucleus, which is largest at the center and reduces exponentially at larger distances. The electron is most likely to be found at a distance from the nucleus equal to the Bohr radius. This function is known as the 1s atomic orbital. For hydrogen (H), an electron in the ground state has energy 3017782103997768000♠−13.6 eV, relative to the ionization threshold. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom.
The exact definition of one second of time since 1997 has been the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest at a temperature of 0 K.[2]
Notes[edit]
Jump up ^ See for example, Cohen, M. (1956). "Appendix A: Proof of non-degeneracy of the ground state". The energy spectrum of the excitations in liquid helium (PDF) (Ph.D.). California Institute of Technology. Published as Feynman, R. P.; Cohen, Michael (1956). "Energy Spectrum of the Excitations in Liquid Helium". Physical Review. 102 (5): 1189. Bibcode:1956PhRv..102.1189F. doi:10.1103/PhysRev.102.1189.
Jump up ^ "Unit of time (second)". SI Brochure. International Bureau of Weights and Measures. Retrieved 2013-12-22.
Bibliography[edit]
Feynman, Richard; Leighton, Robert; Sands, Matthew (1965). "see section 2-5 for energy levels, 19 for the hydrogen atom". The Feynman Lectures on Physics. 3.
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Strain theory (sociology)
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In sociology and criminology, strain theory states that social structures within society may pressure citizens to commit crime. Following on the work of Émile Durkheim, strain theories have been advanced by Robert King Merton (1938), Albert K. Cohen (1955), Richard Cloward, Lloyd Ohlin (1960), Neil Smelser (1963), Robert Agnew (1992), Steven Messner and Richard Rosenfeld (1994).
Contents
[hide]
1 Strain theory
1.1 Merton's theory
2 Derived theories
2.1 General strain theory
2.2 Institutional anomie theory
2.3 Illegitimate opportunities theory
3 Other strain theorists
3.1 Robert Agnew
3.2 Jie Zhang
4 Criticism
5 Studies
6 See also
7 References
8 External links
Strain theory[edit]
Strain theory is a sociology and criminology theory developed in 1938 by Robert K. Merton.[1] The theory states that society puts pressure on individuals to achieve socially accepted goals (such as the American dream) though they lack the means, this leads to strain which may lead the individuals to commit crimes. Examples being selling drugs or becoming involved in prostitution to gain financial security.[1]
Strain may either be:
Structural: this refers to the processes at the societal level which filter down and affect how the individual perceives his or her needs, i.e. if particular social structures are inherently inadequate or there is inadequate regulation, this may change the individual's perceptions as to means and opportunities; or
Individual: this refers to the frictions and pains experienced by an individual as he or she looks for ways to satisfy his or her needs, i.e. if the goals of a society become significant to an individual, actually achieving them may become more important than the means adopted.
Merton's theory[edit]
Robert King Merton was an American sociologist who argued that society can encourage deviance to a large degree. Merton believed that socially accepted goals put pressure on people to conform. People are forced to work within the system or become members of a deviant subculture to achieve the desired goal. Merton's belief became the theory known as Strain Theory. Merton continued on to say when individuals are faced with a gap between their goals (usually finances/money related) and their current status, strain occurs. When faced with strain, people have five ways to adapt:[1]
Conformity: pursuing cultural goals through socially approved means. ("Hopeful poor")
Innovation: using socially unapproved or unconventional means to obtain culturally approved goals. Example: dealing drugs or stealing to achieve financial security. ("surviving poor")
Ritualism: using the same socially approved means to achieve less elusive goals (more modest and humble). ("passive poor")
Retreatism: to reject both the cultural goals and the means to obtain it, then find a way to escape it. ("retreating poor")
Rebellion: to reject the cultural goals and means, then work to replace them. ("resisting poor")
Derived theories[edit]
General strain theory[edit]
Main article: General strain theory
General strain theory (GST) is a sociology and criminology theory developed in 1992 by Robert Agnew. Agnew believed that Merton's theory was too vague in nature and did not account for criminal activity which did not involve financial gain. The core idea of general strain theory is that people who experience strain or stress become distressed or upset which may lead them to commit crime in order to cope. One of the key principle of this theory is emotion as the motivator for crime. The theory was developed to conceptualize the full range of sources in society where strain possibly comes from, which Merton's strain theory does not. The theory also focuses on the perspective of goals for status, expectations and class rather than focusing on money (as Merton's theory does). Examples of General Strain Theory are people who use illegal drugs to make themselves feel better, or a student assaulting his peers to end the harassment they caused.[2][3][4]
GST introduces 3 main sources of strain such as:[5]
Loss of positive stimuli (death of family or friend)
Presentation of negative stimuli (physical and verbal assaults)
The inability to reach a desired goal.
Institutional anomie theory[edit]
Institutional anomie theory (IAT) is a criminology theory developed in 1994 in by Steven Messner and Richard Rosenfeld. The theory proposes that an institutional arrangement with a market, where the market/economy is allowed to operate/dominate without restraints from other social intuitions like family will likely cause criminal behavior. Derived from Merton's Strain Theory, IAT expands on the macro levels of the theory. IAT's focus centers on the criminal influences of varied social institutions, rather than just the economic structure.[6][7]
Illegitimate opportunities theory[edit]
Main article: Illegitimate opportunity
Illegitimate opportunities is a sociology theory developed in 1960 by Richard Cloward and Lloyd Ohlin. The theory states that crimes result from a high number of illegitimate opportunities and not from a lack of legitimate ones. The theory was created from Merton's strain theory to help address juvenile delinquency.[8]
Other strain theorists[edit]
Robert Agnew[edit]
In 1992, Robert Agnew asserted that strain theory could be central in explaining crime and deviance, but that it needed revision so that it was not tied to social class or cultural variables, but re-focused on norms. To this end, Agnew proposed a general strain theory that is neither structural nor interpersonal but rather individual and emotional, paying special attention to an individual's immediate social environment. He argued that an individual's actual or anticipated failure to achieve positively valued goals, actual or anticipated removal of positively valued stimuli, and actual or anticipated presentation of negative stimuli all result in strain.
Anger and frustration confirm negative relationships. The resulting behavior patterns will often be characterized by more than their share of unilateral action because an individual will have a natural desire to avoid unpleasant rejections, and these unilateral actions (especially when antisocial) will further contribute to an individual's alienation from society. If particular rejections are generalized into feelings that the environment is unsupportive,more strongly negative emotions may motivate the individual to engage in crime. This is most likely to be true for younger individuals, and Agnew suggested that research focus on the magnitude, recency, duration, and clustering of such strain-related events to determine whether a person copes with strain in a criminal or conforming manner. Temperament, intelligence, interpersonal skills, self-efficacy, the presence of conventional social support, and the absence of association with antisocial (e.g., criminally inclined) age and status peers are chief among the factors Agnew identified as beneficial.
Jie Zhang[edit]
The strain theory of suicide postulates that suicide is usually preceded by psychological strains. A psychological strain is formed by at least two stresses or pressures, pushing the individual to different directions. A strain can be a consequence of any of the four conflicts: differential values, discrepancy between aspiration and reality, relative deprivation, and lack of coping skills for a crisis. Psychological strains in the form of all the four sources have been tested and supported with a sample of suicide notes in the United States and in rural China through psychological autopsy studies. The strain theory of suicide forms a challenge to the psychiatric model popular among the suicidologists in the world.
The strain theory of suicide is based on the theoretical frameworks established by previous sociologists, e.g. Durkheim (1951), Merton (1957), and Agnew (2006), and preliminary tests have been accomplished with some American (Zhang and Lester 2008) and Chinese data (Zhang 2010; Zhang, Dong, Delprino, and Zhou 2009; Zhang, Wieczorek, Conwell, and Tu 2011).
There could be four types of strain that precede a suicide, and each can be derived from specific sources. A source of strain must consist of two, and at least two, conflicting social facts. If the two social facts are non-contradictory, there would be no strain.
Strain Source 1: Differential Values
When two conflicting social values or beliefs are competing in an individual’s daily life, the person experiences value strain. The two conflicting social facts are competing personal beliefs internalized in the person’s value system. A cult member may experience strain if the mainstream culture and the cult religion are both considered important in the cult member’s daily life. Other examples include the second generation of immigrants in the United States who have to abide by the ethnic culture rules enforced in the family while simultaneously adapting to the American culture with peers and school. In China, rural young women appreciate gender egalitarianism advocated by the communist government, but at the same time, they are trapped in cultural sexual discrimination as traditionally cultivated by Confucianism. Another example that might be found in developing countries is the differential values of traditional collectivism and modern individualism. When the two conflicting values are taken as equally important in a person’s daily life, the person experiences great strain. When one value is more important than the other, there is then little or no strain.
Criticism[edit]
Strain theory has received several criticisms, such as:[9][10]
Strain theory best applies only to the lower class as they struggle with limited resources to obtain their goals.
Strain theory fails to explain white collar crime, the perpetrator of whom have many opportunities to achieve through legal and legitimate means.
Strain theory fails to explain crimes based in gender inequality.
Merton deals with individuals forms of responses instead of group activity which crime involves.
Merton's theory is not very critical of the social structure that he says generate the strains.
Strain theory neglects the inter- and intra-personal aspect of crime.
Strain theory has weak empirical evidence supporting it.
Studies[edit]
Strain theory was tested following its development. Most of these test examined ideal goals such as occupational goals and individual expectations, which would most ideally lead to crimes if not achieved under rule of strain theory. However, most of the research found that this was not the case. An example of these studies was a study done by Travis Hirschi in the 1969. He analyzes a large body of data on delinquency collected in Western Contra Costa County, California that contrast with strain theory.[11] These results and other criticisms lead to the abandonment of strain theory around the 1970s to the 80s.
See also[edit]
Illegitimate opportunity
References[edit]
^ Jump up to: a b c Merton, Robert (1938). "Social Structure and Anomie". American Sociological Review. 3 (5): 672–682. doi:10.2307/2084686.
Jump up ^ Agnew, Robert (2001). "Building on the Foundation of General Strain Theory: Specifying the Types of Strain Most Likely to Lead to Crime and Delinquency". Journal of Research in Crime and Delinquency. 38 (4): 319–361. doi:10.1177/0022427801038004001.
Jump up ^ Agnew, Robert (2014). "General Strain Theory". Journal of Research in Crime and Delinquency: 1892–1900.
Jump up ^ Agnew, Robert (2015). "General Strain Theory and Delinquency". The Handbook of Juvenile Delinquency and Juvenile Justice Krohn/The Handbook of Juvenile Delinquency and Juvenile Justice: 237–256.
Jump up ^ Paternoster, Raymond; Mazerolle, Paul (1994). "General Strain Theory and Delinquency: A Replication and Extension". Journal of Research in Crime and Delinquency. 31 (3): 235–263. doi:10.1177/0022427894031003001.
Jump up ^ Messner, Steven; Rosenfeld, Richard (1997). Crime and the American Dream. Belmont, CA: Wadsworth Pub. Co. |access-date= requires |url= (help)
Jump up ^ Savolainen, Jukka. "Institutional Anomie Theory". Oxford Bibliographies Online Datasets.
Jump up ^ Cloward, Richard; Ohlin, Lloyd (1960). Delinquency and opportunity: a theory of delinquent gangs. Glencoe, IL: Free Press.
Jump up ^ Bernard, T.J. (1984). "Control Criticisms of Strain Theories: An Assessment of Theoretical and Empirical Adequacy". Journal of Research in Crime and Delinquency. 21 (4): 353–372. doi:10.1177/0022427884021004005. |access-date= requires |url= (help)
Jump up ^ Kornhauser, Ruth (1978). Social sources of delinquency: an appraisal of analytic models. Chicago: University of Chicago Press.
Jump up ^ Hirschi, Travis (1969). Causes of Delinquency. Berkeley: University of California.
O'Grady W. (2011). "Crime in Canadian Context." Strain/anomie theory 92-94
Agnew, R (1992). "Foundation for a General Strain Theory". Criminology. 30 (1): 47–87. doi:10.1111/j.1745-9125.1992.tb01093.x.
Agnew, R.; White, H. (1992). "An Empirical Test of General Strain Theory". Criminology. 30 (4): 475–99. doi:10.1111/j.1745-9125.1992.tb01113.x.
Agnew, R. (1997). "The Nature and Determinants of Strain: Another Look at Durkheim and Merton." pp. 27–51 in The Future of Anomie Theory, edited by R. Agnew and N. Passas. Boston: Northeastern University Press.
Agnew, R. (2009). "Revitalizing Merton: General Strain Theory." Advances in Criminological Theory: The Origins of American Criminology, Volume 16, edited by F.T. Cullen, F. Adler, C.L. Johnson, and A.J. Meyer. New Brunswick, NJ: Transaction.
Akers, R. (2000). Criminological Theories: Introduction, Evaluation, and Application. Los Angeles: Roxbury.
Cloward, R (1959). "Illegitimate Means, Anomie and Deviant Behavior". American Sociological Review. 24 (2): 164–76. doi:10.2307/2089427.
Cloward, R. & Ohlin, L. (1960). Delinquency and Opportunity. NY: Free Press.
Cohen, A. (1955). Delinquent Boys. NY: Free Press.
Cohen, A (1965). "The Sociology of the Deviant Act: Anomie Theory and Beyond". American Sociological Review. 30: 5–14. doi:10.2307/2091770.
Cohen, A (1977). "The Concept of Criminal Organization". British Journal of Criminology. 17: 97–111. doi:10.1093/oxfordjournals.bjc.a046820.
Dubin, R (1959). "Deviant Behavior and Social Structure: Continuities in Social Theory". American Sociological Review. 24: 147–163. doi:10.2307/2089426.
Durkheim, E. (1897/1997). Suicide. NY: Free Press.
Featherstone, R. & Deflem, M. (2003). "Anomie and Strain: Context and Consequences of Merton's Two Theories." Sociological Inquiry 73(4):471-489.
Hirschi, T. (1969). Causes of Delinquency. Berkeley: University of California Press.
Marwah, Sanjay, and Mathieu Deflem. 2006. ”Revisiting Merton: Continuities in the Theory of Anomie-and-Opportunity-Structures." pp. 57–76 in Sociological Theory and Criminological Research: Views from Europe and the United States, ed. M. Deflem. Amsterdam: Elsevier/JAI Press.
Messner, S & Rosenfeld, R. (1994). Crime and the American Dream. Belmont: Wadsworth.
Polk, K (1969). "Class, Strain and Rebellion Among Adolescents". Social Problems. 17: 214–24. doi:10.2307/799867.
Polk, K., & Schafer, W. (eds.). (1972). Schools and Delinquency. Englewood Cliffs, NJ: Prentice-Hall.
Agnew, Robert. 2006. "General Strain Theory: Current Status and Directions for Further Research." pp. 101–123 in Taking Stock: The Status of Criminological Theory-Advances in Criminological Theory, edited by F. T. Cullen, J. P. Wright, and K. Blevins. New Brunswick, NJ: Transaction.
Durkheim, Emile. 1951. Suicide: A Study in Sociology. New York: Free Press (Original work published in 1897).
IOM, (Institute of Medicine). 2002. Reducing suicide: An American imperative. Washington, D.C.: National Academy Press.
Mann, J.J.; Waternaux, C.; Haas, G.L.; Malone, K.M. (1999). "Toward a clinical model of suicidal behavior in psychiatric patients". American Journal of Psychiatry. 156: 181–189. doi:10.1176/ajp.156.2.181.
Merton, R.K. 1957. Social Theory and Social Structure, rev. ed. New York: Free Press.
NIMH. 2003. Research on Reduction and Prevention of Suicidality: National Institute of Mental Health.
Phillips, Michael R; Yang, Gonghuan; Zhang, Yanping; Wang, L.; Ji, H.; Zhou, M. (2002). "Risk factors for suicide in China: a national case-control psychological autopsy study". The Lancet. 360: 1728–1736. doi:10.1016/s0140-6736(02)11681-3.
Spitzer, R.L., J.B.W. Williams, M. Gibbon, and A.B. First. 1988. Instruction Manual for the Structured Clinical Interview for DSM-III-R (SCID, 6/1/88 Revision). New York: Biometrics Research Department, New York State Psychiatric Institute.
Zhang, Jie (2010). "Marriage and Suicide among Chinese Rural Young Women". Social Forces. 89: 311–326. doi:10.1353/sof.2010.0065.
Zhang, Jie; Dong, Nini; Delprino, Robert; Zhou, Li (2009). "Psychological Strains Found From In-Depth Interviews With 105 Chinese Rural Youth Suicides". Archives of Suicide Research. 13: 185–194. doi:10.1080/13811110902835155.
Zhang, Jie and Shenghua Jin. 1998. "Interpersonal relations and suicide ideation in China." Genetic, Social, and General Psychology Monographs 124:79-94.
Zhang, Jie; Lester, David (2008). "Psychological Tensions Found in Suicide Notes: A Test for the Strain Theory of Suicide". Archives of Suicide Research. 12: 67–73. doi:10.1080/13811110701800962.
Zhang, Jie; Wieczorek, William F.; Conwell, Yeates; Ming Tu, Xin (2011). "Psychological strains and youth suicide in rural China". Social Science & Medicine. 72: 2003–2010. doi:10.1016/j.socscimed.2011.03.048.
Zhang, Jie. 2000. "Gender differences in athletic performance and their implications in gender ratios of suicide: A comparison between the USA and China." Omega: Journal of Death and Dying 41:117-123.
External links[edit]
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Leg bone
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