What is Schrodinger's equation?

Answer 1

The Schrödinger wave equation shows the interactions between particles and potential fields (i.e., electrons within atoms) by describing the behavior of such a system. Elaborating a little more, a particle is described by what is called a wavefunction. This wavefunction has a space (x,y,z) and time (t) dependency and is continuous, finite and single valued. Therefore the Schrödinger wave partial differential equation shows how the wavefunction of a system behaves over time.

Answer 2

Assumptions: * 1- The total energy E of a particle is

: :: This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, , and the potential energy V. pis the momentum of the particle or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well. Note that the energy E and momentum pappear in the following two relations: * 2- Einstein's light quanta hypothesis of 1905: : :: ::: where the frequency f of the quanta of radiation (photons) are related by Planck's constant h. * 3- The de Broglie hypothesis of 1924: : :: ::: where is the wavelength of the wave. This hypothesis also requires: * 4- The association of a wave (with wavefunction ψ) with any particle. Combining the above assumptions yields Schrödinger's equation: Expressing frequency f in terms of angular frequency and wavelength in terms of wavenumber , with we get: : :: and : :: where we have expressed p and k as vectors. Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor: : :: and to realize that since : :: then : :: and similarly since: : :: then : :: and hence: : :: so that, again for a plane wave, he obtained: : :: And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V: : :: Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment for the Lyman, Balmer, Paschen and Brackett series, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a paper in 1926.[2] Assumptions: * 1- The total energy E of a particle is

: :: This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, , and the potential energy V. pis the momentum of the particle or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well. Note that the energy E and momentum pappear in the following two relations: * 2- Einstein's light quanta hypothesis of 1905: : :: ::: where the frequency f of the quanta of radiation (photons) are related by Planck's constant h. * 3- The de Broglie hypothesis of 1924: : :: ::: where is the wavelength of the wave. This hypothesis also requires: * 4- The association of a wave (with wavefunction ψ) with any particle. Combining the above assumptions yields Schrödinger's equation: Expressing frequency f in terms of angular frequency and wavelength in terms of wavenumber , with we get: : :: and : :: where we have expressed p and k as vectors. Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor: : :: and to realize that since : :: then : :: and similarly since: : :: then : :: and hence: : :: so that, again for a plane wave, he obtained: : :: And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V: : :: Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment for the Lyman, Balmer, Paschen and Brackett series, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a paper in 1926.[2] Assumptions: * 1- The total energy E of a particle is

: :: This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, , and the potential energy V. pis the momentum of the particle or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well. Note that the energy E and momentum pappear in the following two relations: * 2- Einstein's light quanta hypothesis of 1905: : :: ::: where the frequency f of the quanta of radiation (photons) are related by Planck's constant h. * 3- The de Broglie hypothesis of 1924: : :: ::: where is the wavelength of the wave. This hypothesis also requires: * 4- The association of a wave (with wavefunction ψ) with any particle. Combining the above assumptions yields Schrödinger's equation: Expressing frequency f in terms of angular frequency and wavelength in terms of wavenumber , with we get: : :: and : :: where we have expressed p and k as vectors. Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor: : :: and to realize that since : :: then : :: and similarly since: : :: then : :: and hence: : :: so that, again for a plane wave, he obtained: : :: And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V: : :: Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment for the Lyman, Balmer, Paschen and Brackett series, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a paper in 1926.[2]

Answer 3

The is Schrodinger equation comes in two main flavors, time dependent and time independent. This the time independent equation allows us to determine the spatial wave equation in any problem.

Time independent;

[(-hbar^2)/2*m]((d/dx)^2)*(psi)+U*(psi)=E*(psi)

where hbar=h/(2pi) ; h is defined as planck's constant, m is the mass, (d/dx)^2 is the second spacial derivative, psi it the greek letter which is the time independent wave function, U is the potential, E is the energy.

Time dependent:

i(hbar)(d/dt)*Psi=[(-hbar^2)/2*m]((d/dx)^2)*(Psi)+U*(Psi)

where i=sqrt(-1), (d/dt) is the time derivative, Psi is the space and time dependent wave function.

NOTE!!! psi is not the same as Psi in my above notation.

psi is only a function of x

Psi is a function of x and t

Answer 4

I think you are meaning picture metaphorically, not literally a picture. When Schrödinger formulated his wave equation he drew from a classical foundation. Total energy is equal to the kinetic energy plus the potential energy. Classically kinetic energy is expressed as p2/(2m), where p is momentum and m is the mass. The potential can be whatever you want it to be so we will just call it V, then finally lets call the total energy E. We put this all together to get p2/(2m)+V=E but in terms of operators p=[i*h/(2*pi)]*(d2/dqi2), where i is the square root of negative one, h is Planck's constant, pi is 3.141529, and (d2/dqi2) is the second order differential with respect to space in generalized coordinates. So putting p back in our equation for energy we get [- h2/(4*pi*m)]*(d2/dqi2)+V=E. Now comes the sort of complex part. In general the total energy of a system is defined by the Hamiltonian, lets call is H. Now, E is the eigenvalue of the Hamiltonian operator. Meaning that when H operates on some state it is measuring the total energy of that state. In quantum mechanics H=[i*h/(2*pi)]*(d/dt), where (d\dt) is the first order differential with respect to time. Now putting this all together we get [i*h/(2*pi)]*(d/dt)=[- h2/(4*pi*m)]*(d2/dqi2)+V. That, to me, is Schrodinger's "picture" of the wave equation.

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Sorry it is a bit lengthy but I wanted to be thorough. Also, if you feel that I did not address your question properly then please leave a message on my wall.

Answer 5

(SKIP FOR EXAMPLE)

The solution to Schrodinger's equation depends on the potential energy of the particle. Potential energy is, in general, a function of the position; thus, each situation requires solving Schrodinger's equation for that particular potential. What is to be obtained in solving Shcrodinger's equation is the wave function for the object in mind. This wave function allows us to determine the probability of observing a particle at some position/time (summed over all positions must be 1 since the particle must be observed somewhere), the energy of the particle, the momentum, etc.

Once you know the potential you are dealing with, Schrodinger's equation is solved using techniques of partial differential equations. If one has studied single variable differential equations and is willing to accept a hand waving argument one can solve Schrodinger's equation in one dimension. The equation itself isn't a particularly difficult differntial equation, but partial differential equations in general are difficult to understand. I suggest reading Griffith's book on quantum mechanics if you are truly interested.

(EXAMPLE: If this is too much, read the paragraphs above.)

If considering a potential well (0 potential inside the well and infinite potential outside, which has a broad range of applications), centered at the origin, for a particle, and allowing me to make said hand waving argument, a solution may be derived. For convention I will take (f*) to be the time derivative of f and f' to be the x derivative (f'' is the second x derivative) of f. Also I will take h_ to be the reduced planck's constant (simply planck's constant diveded by 2Pi) and i to be the imaginary unit. Schrodinger's equation with constant potential and our new convention becomes:

(1): -[(h_2)/2m] f''+V f=i h_ (f*)

In equation (1) f is the wave function, which is what we are solving for. Applying separation of variables yields a differential equation in F and T, where F is a function of position, T is a function of time, and the product of F and T is our wavefunction, i.e., f=F*T. The equations for F and T are:

(2): (T*)=-[(i E)/h_]T

(3): -[(h_2)/2m] F''=[E-V] F

where the constant E is the energy the particle has. This E is a constant from separation of variables. If I did not know it would work out to be the energy of the particle I could have used another variable only to find out later that it must be the energy of the particle. Equation (3) tells us that the second derivative of F is proportional to F itself. Equation (2) has the solution (if you don't understand this part you need to take differential equations):

(4): T=e^(-[i E/h_]t)

where t is time. Any multiple of this equation is a solution, but the same happens for equation (3), thus we let an arbitrary constant be assigned later rather than in equation (4). Solving equation (3) yields:

(5): F=A cos(-[2m/(h_2)][E-V]x)

A is an arbitrary constant since any multiple of the solution is itself a solution (superposition principle). To get rid of this constant we impose that F be normalized, i.e., that the particle must be found somewhere. Inside the well V is 0, thus the wave function is sinusoidal. Since we know it must be zero at the walls we can impose constraints on E. This quantization of energy agrees with observations of quantum energies. Outside the well V is infinite, thus the wave function is 0 since we do not expect to observe our particle here. Our constraints on E and normalization constant are:

(6): E=-[h_2/2m][n Pi/L] (where n is an integer and L is the length of the well)

(7): A=1/Sqrt[L]

Putting all this together, the wave equation for our particle is:

(6): f=F*T=e^(-[2m/(h_2)][E-V]x) e^(-[i E/h_]t)/Sqrt[L]

This whole thing is a convoluted mess, and to make things worse, this is one of the simplest cases of Schrodinger's equation. The point, though, is that it is not impossible, and also, that solving the equation depends on the potential you are given. Schrodinger's equation applies to all things. The difficulty is in knowing the potential function, and furthermore, solving the equation for real world potentials.

Answer 6

schrodinger wave equation for hydrogen atom solution

Answer 7

Shroedinger's wave equation is used to find the wavefunction of a particle, telling us everything we can possibly know about that particle at that time.

Answer 8

Schrodinger equation provides relation between particles involved in the event, its energy and wave function (its derivatives).