Value of Wave Functions - Coursework Example

Wave function: Nonlinear waves and quasilinear equations Name Date Introduction A wave function is described as a variable quantity that mathematically describes the wave characteristics of a particle or mathematically describe a system quantum state. A wave function of a particle has value and the value at a given space and time is related to the possibility of that particle present in that space at that time. It is normally represented by a Greek letter psi Ψ. It is a complex probability amplitude and is used to derive the probability for the expected outcome for the measurements made on the system (Dilip, 2016). Furthermore, it is a degrees of freedom function that is equivalent to certain maximum set of moving observables. After selecting wave form function representation, the function may then be derived from quantum state. The choice of the commuting degrees of freedom and the consequent domain of wave function are not distinctive such as a function representing all the coordinates of the position of the particles on the space or in some occasions taken as the momentum of all the particles on the momentum space, they are both related by Fourier transform. There are certain particle that have nonzero spin such as photons and electrons and wave function for the particles include discrete degree of freedom and spin as an intrinsic (Binney & Skinner, 2013). A system can have an internal and external degree of freedom, when it is having an internal degree of freedom, the wave function at every point in uninterrupted degrees of freedom allocates a complex number in all values of the discrete degrees of freedom such as the spin z-component. The wave functions values may be added together then multiplied by complex numbers resulting into different wave functions and produce Hilbert space according to superposition principle of quantum mechanics (Binney & Skinner, 2013). The product from the two wave’s functions has an inner product which is the degree of the overlap between the inner product and the equivalent physical states is used in the fundamental probable quantum mechanics interpretation (Binney & Skinner, 2013). The concept of wave function was first presented in Schrodinger equation. Wave functions have been evolving with time and its evolution is determined by Schrödinger equation. The function of waves behaves qualitatively in a similar way to other waves like water waves, the reason is that Schrödinger equation is a wave equation, which produce the name wave function and produce a wave–particle duality (Schneider, 1998). Conversely, quantum physics describe wave function as a form of physical phenomenon and it can be interpreted in different ways and it differs from the interpretation of classical mechanical waves. The wave function square Ψ2 is important because it provides the probability of finding the particle described by specific wave function at a given position and time, because the answer from the wave function is normally a complex number (Schneider, 1998). In non-relativistic quantum physics, Born’s did some statistical interpretation of quantum mechanics and he found that the modulus square of the wave function | ψ |2 is a real number which is taken as a probability density of the particles identified at a specified point or which possess a particular momentum at given time and probably have certain values for distinct degree of freedom (Schneider, 1998). Its integral quantity compared to other degrees of freedom systems should be 1 according to the probability interpretation. A full wave function must satisfy certain requirements that are known as normalization condition. A wave function phase and relative magnitude can only be measured because wave function is complexed valued. The formula of finding wave function is  Whereby, i is an imaginary number Ψ (x, t) is wave function ħ is reduced Planck constant. t is time x is position in space and Ĥ is mathematical object called Hamilton operator. The properties of wave functions are; they contain all measurable information about a particle; Ψ+Ψ summed over all space is equivalent to 1, i.e. if the particle exist there, the probability of finding it there is 1; it is continuous; permit calculation of energy through Schrödinger equation; establish probability distribution in 3 dimensions; allow calculation of effective average value of a variable; and free particle is a sign wave. There are two types of waves: linear and nonlinear waves. Nonlinear waves are defined by nonlinear equations thus, principle of superposition will not apply, which makes this equation difficult to analyze mathematically and the analytical technique for solving this equations has never been developed. Every particular wave equation has to be taken as an individual. There is no general method of finding solutions to nonlinear PDEs. The trial way of using different methods are applied, once a method has been applied it is then investigated whether or not it satisfies particular PDE. The following are a few solution methods used in solving nonlinear wave equations, generalized separation of variables method, Differential constraints method, Group analysis methods, Hirota's bilinear method, Hodograph transformation method, Inverse scattering transform (IST) method, Backlund transformation and many others (Johnson, 1997). Waves are time evolutions phenomena’s that can be modelled mathematically by use of partial differential equations (PDE), that have dependent variable u(x,t) that represents the wave value, independent variable which represent time t, and spatial independent variables x ∈ Rn where n is equal to 1, 2, 3, 4,.. (Johnson, 1997). General PDE of order m can be written as F(x, δau(x); | α | ≤ m) = 0, x € Ω. Whereby Ω ⊂ Rd and the notations means that F depends on u and all of its derivatives of order less than or equal to m (Johnson, 1997). It is assumed that u is a real value. Nonlinear equations are divided into different classes depending on their linearity, they are semi linear if it has a form Quasilinear if it has a form And it is said to be fully linear if it depends nonlinearity on highest derivatives. Examples of nonlinear wave equations are Burgers equation: ut + uux – auxx =0 and its solution is u(x, t) = 2ak [1−tanh k (x−Vt)]. Whereby α is arbitrary constant, K is the wave number, V is the velocity, and Korteweg-de Vries is given by  and its solution is  for a travelling wave whereby sech stands for hyperbolic secant, c is phase speed. Quasilinear equations are equations that are linear in their derivatives, this equations normally occur in applications for example in fluid mechanics. Quasilinear equations can be understood better by studying Cauchy problem for quasilinear equation. We start with the equation xt +c(x, t, u) ux=f (x, t, u), x ∈ R, t>0 eq. 1 Whereby the initial condition has the form u(x, 0) =ø(x), x ∈ R eq. 2 Curves are defined by differential equation  on curves is no longer a constant but instead  the third and fourth equations are known as characteristic system associated with the first equation. The initial data may be represented in any form for example x= Ɛ, u= ø (Ɛ), on t= 0 whereby Ɛ is a parameter representing arbitrary value on x axis. An example in which this equations can be applied is in solving initial value problems (IVP). Ut + uux = -u, x ∈ R, t>0  Characteristics system is and initial conditions gives x= Ɛ, u= -  on t= 0 General solution of characteristic system is  Applying the initial data gives Ɛ= -c1 + c2 and – = c1 hence  therefore Ɛ = Ɛ(x, t) =  and so u(x, t) =  Quasilinear equations such as eq.1 can be solved in form of particular arbitrary function. To come up with general solution a number of definition and observation are looked at.  Expression is known as first integral of characteristic system  and  in situation where  on solutions to the two equations. That is  for all t and in some situations I, so if  is a solution to the equations. If the full derivative of the above equation in respect to t and applying chain rule it gives this equation  whereby every term in the equation is evaluated at (David, 2013). By use of chain rule the partial derivatives of u can be calculated and it gives  this curve  lies on that surface since (David, 2013). The surface is a result of the PDE eq.1. Another example of deriving a solution using PDE of wave function is by considering the equation . eq.1 The characteristic system is given by  = eq.2 Equation one gives 1st integral  and the 2nd equation then becomes  when the variables are separated and then integrated gives another first integral that is (David, 2013). Hence the general solution for eq.1 is  in which h is arbitrary function. References David, L. J. (2013). Applied mathematics. 4th Ed. New York: J. Wiley. Binney, J., & Skinner, D. (2013). The physics of quantum mechanics. Oxford: Oxford University Press Dilip D. J., (2016). Neo-Classical Physics or Quantum Mechanics? A New Theory of Physics, Educreation Publishing Hörmander, L. (1997). Lectures on nonlinear hyperbolic differential equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin John, F. (1991). Partial differential equations, vol. 1 of Applied Mathematical Sciences, Springer-Verlag. New York. Johnson, R. S. (1997). A modern introduction to the mathematical theory of water waves. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press. Linares, F. and Ponce, G. (2009). Introduction to nonlinear dispersive equations, Universitext. New York: Springer. Schneider, G. (1998). Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation. J. Differ. Equations, 147, pp. 333–354. Draft Wave function: - Nonlinear waves and quasilinear equations Name Date Introduction A wave function is described as a variable quantity that mathematically describes the wave characteristics of a particle or mathematically describe a system quantum state. A wave function of a particle has value and the value at a given space and time is related to the possibility of that particle present in that space at that time. It is normally represented by a Greek letter psi Ψ. It is a complex probability amplitude and is used to derive the probability for the expected outcome for the measurements made on the system (Dilip, 2016). Furthermore, it is a degrees of freedom function that is equivalent to certain maximum set of moving observables. After selecting wave form function representation, the function may then be derived from quantum state. The choice of the commuting degrees of freedom and the consequent domain of wave function are not distinctive such as a function representing all the coordinates of the position of the particles on the space or in some occasions taken as the momentum of all the particles on the momentum space, they are both related by Fourier transform. There are certain particle that have nonzero spin such as photons and electrons and wave function for the particles include discrete degree of freedom and spin as an intrinsic (Binney & Skinner, 2013). A system can have an internal and external degree of freedom, when it is having an internal degree of freedom, the wave function at every point in uninterrupted degrees of freedom allocates a complex number in all values of the discrete degrees of freedom such as the spin z-component. The wave functions values may be added together then multiplied by complex numbers resulting into different wave functions and produce Hilbert space according to superposition principle of quantum mechanics (Binney & Skinner, 2013). The product from the two wave’s functions has an inner product which is the degree of the overlap between the inner product and the equivalent physical states is used in the fundamental probable quantum mechanics interpretation (Binney & Skinner, 2013). The concept of wave function was first presented in Schrodinger equation. Wave functions have been evolving with time and its evolution is determined by Schrödinger equation. The function of waves behaves qualitatively in a similar way to other waves like water waves, the reason is that Schrödinger equation is a wave equation, which produce the name wave function and produce a wave–particle duality (Schneider, 1998). Conversely, quantum physics describe wave function as a form of physical phenomenon and it can be interpreted indifferent ways and it differs from the interpretation of classical mechanical waves. The wave function squareΨ2 is important because it provides the probability of finding the particle described by specific wave function at a given position and time, because the answer from the wave function is normally a complex number (Schneider, 1998). In non-relativistic quantum physics, Born’s did some statistical interpretation of quantum mechanics and he found that the modulus square of the wave function | ψ |2 is a real number which is taken as a probability density of the particles identified at a specified point or which possess a particular momentum at given time and probably have certain values for distinct degree of freedom (Schneider, 1998). Its integral quantity compared to other degrees of freedom systems should be 1 according to the probability interpretation. A full wave function must satisfy certain requirements that are known as normalization condition. A wave function phase and relative magnitude can only be measured because wave function is complexed valued. The formula of finding wave function is ℎ Wave function: Nonlinear waves and quasilinear equations Name Date Introduction A wave function is described as a variable quantity that mathematically describes the wave characteristics of a particle or mathematically describe a system quantum state. A wave function of a particle has value and the value at a given space and time is related to the possibility of that particle present in that space at that time. It is normally represented by a Greek letter psi Ψ. It is a complex probability amplitude and is used to derive the probability for the expected outcome for the measurements made on the system (Dilip, 2016). Furthermore, it is a degrees of freedom function that is equivalent to certain maximum set of moving observables. After selecting wave form function representation, the function may then be derived from quantum state. The choice of the commuting degrees of freedom and the consequent domain of wave function are not distinctive such as a function representing all the coordinates of the position of the particles on the space or in some occasions taken as the momentum of all the particles on the momentum space, they are both related by Fourier transform. There are certain particle that have nonzero spin such as photons and electrons and wave function for the particles include discrete degree of freedom and spin as an intrinsic (Binney & Skinner, 2013). A system can have an internal and external degree of freedom, when it is having an internal degree of freedom, the wave function at every point in uninterrupted degrees of freedom allocates a complex number in all values of the discrete degrees of freedom such as the spin z-component. The wave functions values may be added together then multiplied by complex numbers resulting into different wave functions and produce Hilbert space according to superposition principle of quantum mechanics (Binney & Skinner, 2013). The product from the two wave’s functions has an inner product which is the degree of the overlap between the inner product and the equivalent physical states is used in the fundamental probable quantum mechanics interpretation (Binney & Skinner, 2013). The concept of wave function was first presented in Schrodinger equation. Wave functions have been evolving with time and its evolution is determined by Schrödinger equation. The function of waves behaves qualitatively in a similar way to other waves like water waves, the reason is that Schrödinger equation is a wave equation, which produce the name wave function and produce a wave–particle duality (Schneider, 1998). Conversely, quantum physics describe wave function as a form of physical phenomenon and it can be interpreted in different ways and it differs from the interpretation of classical mechanical waves. The wave function square Ψ2 is important because it provides the probability of finding the particle described by specific wave function at a given position and time, because the answer from the wave function is normally a complex number (Schneider, 1998). In non-relativistic quantum physics, Born’s did some statistical interpretation of quantum mechanics and he found that the modulus square of the wave function | ψ |2 is a real number which is taken as a probability density of the particles identified at a specified point or which possess a particular momentum at given time and probably have certain values for distinct degree of freedom (Schneider, 1998). Its integral quantity compared to other degrees of freedom systems should be 1 according to the probability interpretation. A full wave function must satisfy certain requirements that are known as normalization condition. A wave function phase and relative magnitude can only be measured because wave function is 11 12 8 7 complexed valued. The formula of finding wave function is ℎ Wave function: Nonlinear waves and quasilinear equations Name Date Introduction A wave function is described as a variable quantity that mathematically describes the wave characteristics of a particle or mathematically describe a system quantum state. A wave function of a particle has value and the value at a given space and time is related to the possibility of that particle present in that space at that time. It is normally represented by a Greek letter psi Ψ. It is a complex probability amplitude and is used to derive the probability for the expected outcome for the measurements made on the system (Dilip, 2016). Furthermore, it is a degrees of freedom function that is equivalent to certain maximum set of moving observables. After selecting wave form function representation, the function may then be derived from quantum state. The choice of the commuting degrees of freedom and the consequent domain of wave function are not distinctive such as a function representing all the coordinates of the position of the particles on the space or in some occasions taken as the momentum of all the particles on the momentum space, they are both related by Fourier transform. There are certain particle that have nonzero spin such as photons and electrons and wave function for the particles include discrete degree of freedom and spin as an intrinsic (Binney & Skinner, 2013). A system can have an internal and external degree of freedom, when it is having an internal degree of freedom, the wave function at every point in uninterrupted degrees of freedom allocates a complex number in all values of the discrete degrees of freedom such as the spin z-component. The wave functions values may be added together then multiplied by complex numbers resulting into different wave functions and produce Hilbert space according to superposition principle of quantum mechanics (Binney & Skinner, 2013). The product from the two wave’s functions has an inner product which is the degree of the overlap between the inner product and the equivalent physical states is used in the fundamental probable quantum mechanics interpretation (Binney & Skinner, 2013). The concept of wave function was first presented in Schrodinger equation. Wave functions have been evolving with time and its evolution is determined by Schrödinger equation. The function of waves behaves qualitatively in a similar way to other waves like water waves, the reason is that Schrödinger equation is a wave equation, which produce the name wave function and produce a wave–particle duality (Schneider, 1998). Conversely, quantum physics describe wave function as a form of physical phenomenon and it can be interpreted in different ways and it differs from the interpretation of classical mechanical waves. The wave function square Ψ2 is important because it provides the probability of finding the particle described by specific wave function at a given position and time, because the answer from the wave function is normally a complex number (Schneider, 1998). In non-relativistic quantum physics, Born’s did some statistical interpretation of quantum mechanics and he found that the modulus square of the wave function | ψ |2 is a real number which is taken as a probability density of the particles identified at a specified point or which possess a particular momentum at given time and probably have certain values for distinct degree of freedom (Schneider, 1998). Its integral quantity compared to other degrees of freedom systems should be 1 according to the probability interpretation. A full wave function must satisfy certain requirements that are known as normalization condition. A wave function phase and relative magnitude can only be measured because wave function is 11 12 8 7 complexed valued. The formula of finding wave function is ℎ Read More

The concept of wave function was first presented in Schrodinger equation. Wave functions have been evolving with time and its evolution is determined by Schrödinger equation. The function of waves behaves qualitatively in a similar way to other waves like water waves, the reason is that Schrödinger equation is a wave equation, which produce the name wave function and produce a wave–particle duality (Schneider, 1998). Conversely, quantum physics describe wave function as a form of physical phenomenon and it can be interpreted in different ways and it differs from the interpretation of classical mechanical waves.

The wave function square Ψ2 is important because it provides the probability of finding the particle described by specific wave function at a given position and time, because the answer from the wave function is normally a complex number (Schneider, 1998). In non-relativistic quantum physics, Born’s did some statistical interpretation of quantum mechanics and he found that the modulus square of the wave function | ψ |2 is a real number which is taken as a probability density of the particles identified at a specified point or which possess a particular momentum at given time and probably have certain values for distinct degree of freedom (Schneider, 1998).

Its integral quantity compared to other degrees of freedom systems should be 1 according to the probability interpretation. A full wave function must satisfy certain requirements that are known as normalization condition. A wave function phase and relative magnitude can only be measured because wave function is complexed valued. The formula of finding wave function is  Whereby, i is an imaginary number Ψ (x, t) is wave function ħ is reduced Planck constant. t is time x is position in space and Ĥ is mathematical object called Hamilton operator.

The properties of wave functions are; they contain all measurable information about a particle; Ψ+Ψ summed over all space is equivalent to 1, i.e. if the particle exist there, the probability of finding it there is 1; it is continuous; permit calculation of energy through Schrödinger equation; establish probability distribution in 3 dimensions; allow calculation of effective average value of a variable; and free particle is a sign wave. There are two types of waves: linear and nonlinear waves.

Nonlinear waves are defined by nonlinear equations thus, principle of superposition will not apply, which makes this equation difficult to analyze mathematically and the analytical technique for solving this equations has never been developed. Every particular wave equation has to be taken as an individual. There is no general method of finding solutions to nonlinear PDEs. The trial way of using different methods are applied, once a method has been applied it is then investigated whether or not it satisfies particular PDE.

The following are a few solution methods used in solving nonlinear wave equations, generalized separation of variables method, Differential constraints method, Group analysis methods, Hirota's bilinear method, Hodograph transformation method, Inverse scattering transform (IST) method, Backlund transformation and many others (Johnson, 1997). Waves are time evolutions phenomena’s that can be modelled mathematically by use of partial differential equations (PDE), that have dependent variable u(x,t) that represents the wave value, independent variable which represent time t, and spatial independent variables x ∈ Rn where n is equal to 1, 2, 3, 4,.. (Johnson, 1997).

General PDE of order m can be written as F(x, δau(x); | α | ≤ m) = 0, x € Ω. Whereby Ω ⊂ Rd and the notations means that F depends on u and all of its derivatives of order less than or equal to m (Johnson, 1997). It is assumed that u is a real value. Nonlinear equations are divided into different classes depending on their linearity, they are semi linear if it has a form Quasilinear if it has a form And it is said to be fully linear if it depends nonlinearity on highest derivatives.

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