What is Schrodinger equation: Definition and 564 Discussions

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian.
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".

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  1. G

    Separation of Variables in Spherical Schrodinger Equation

    The normalization condition is: ∫|ψ|^{2}d^{3}r=1 In spherical coordinates: d^{3}r=r^{2}sinθdrdθd\phi Separating variables: ∫|ψ|^{2}r^{2}sinθdrdθd\phi=∫|R|^{2}r^{2}dr∫|Y|^{2}sinθdθd\phi=1 The next step is the part I don't understand. It says: ∫^{∞}_{0}|R|^{2}r^{2}dr=1 and...
  2. S

    Schrodinger Equation in 3 dimension

    How can i solve Schrodinger equation in 3dimension i want to know how can i deduce every equation ? and how can i find equation of spherical harmonic and radial equation ? i need to understand this proof
  3. H

    Imaginary numbers and the real part of the Schrodinger Equation

    At the moment I am studying the Schrodinger equation using this resource. In a 1D solution (sec 3.1 in the paper) they show that a wave function can be expressed as \Psi(x,t)=\sqrt{2}e^{-iE_{n_x}t}\sin (n_x\pi x) where n_x is the quantum number. And they show the real part of the solution in...
  4. T

    Why i in Schrodinger Equation

    Why "i" in Schrodinger Equation Schrodinger Equation is "i*h*dphi/dt=H*phi" That is to say that the change in the state is proportional to a linear tranformation of the actual state (I understood the logic behind that), H is hermitian and that means that its eingenvalues are real (right?)...
  5. C

    Solving a time independent Schrodinger equation

    Homework Statement Hello! I am currently stuck with a time independent Schrodinger equation where the potential "V(x)" is hyperbolic in nature. I was wondering if anyone could give me a hint as to how I should approach this problem in order to get an analytical solution (without using...
  6. M

    The radial schrodinger equation

    Hi, I'm doing a homework problem in my modern physics class and I'm stuck at a point. The question is "Show that the radial probability density of the 1s level in hydrogen has its maximum value at r = a0, where a0 is the Bohr radius" I know that the radial schrodinger equation will...
  7. A

    Does schrodinger equation give complete description of electron?

    is schrodinger equation capable of explaining all the properties of electron dynamics or are there any anomaly..? THANKS IN ADVANCE
  8. R

    Schrodinger equation of particle in box

    hi all please help me... I'm learning schrodinger equation of a particle in a 1-dimensional box. I read a quantum mechanics book written by A. C Phillips. the wavefuction is ψ (x,t)= N sin (kx) e-iEt/hbar but when I compared to what I read from a modern physics book written by Beisser. the...
  9. S

    Schrodinger equation for one dimensional square well

    Homework Statement the question as well as the hint is shown in the 3 attachments Homework Equations The Attempt at a Solution i know how to normalize an equation, however i do not understand what the hint is saying, or how to do these integrals, any guidance would be greatly...
  10. S

    Schrodinger equation, normalizing

    Homework Statement Consider the infinite well, a particle with mass m in the potential V(x) = \begin{cases} 0, & 0 < x < a,\\ \infty, & \text{otherwise,} \end{cases}, At t = 0 the particle is in the state: \Psi(x,0) = B \left[\sin{\left(\frac{l \pi}{a}x\right)} +...
  11. N

    Looking for: Schrödinger equation in terms of phase and modulus

    Hello, I can't seem to find a reference formulating the Schrödinger equation as a set of two differential equations in terms of the modulus |\psi| and the phase S instead as one diff. eq. in terms of the complex wavefunction \psi = |\psi|e^{i S}. Can anyone show me the way?
  12. T

    Solving the Schrodinger Equation for A and l

    I have the following Schrodinger equation: i* (h-bar) * partial derivative of ψ(x,t) w.r.t time = [(m*w^2 / 2) * x^2 * ψ(x,t) ] - (1/2m) * (h-bar)^2 * (laplacian of ψ(x,t))] m>0 is the mass w is a positive constant Assume that the ground state...
  13. H

    Why isn’t time-dependent Schrodinger equation separable?

    Greetings everyone, I haven’t done any quantum in a while, and was reviewing my textbook, Griffiths Ed. 1. The form of the Schrodinger equation I’m using is: i\hbar\partial\Psi/\partialt = -\hbar2/2m * \partial2\Psi/\partialx2 + V\Psi The book says if V is a function of x only, then the...
  14. C

    Schrodinger Equation and wavefunctions

    not all functions are wavefunctions. For functions to be wavefunctions they have to obey a series of "rules". Now, my question is: there are many functions, which obey these rules which aren't eigenfunctions of the hamiltonian, thereby meaning that they don't obey the Schrodinger Equation...
  15. A

    Applying the FTC to an arbitrary solution of the Schrodinger equation

    Suppose you've got a function \psi(t) that satisfies i\dot \psi = H \psi for some self-adjoint Hamiltonian H. I'd like to apply the fundamental theorem of calculus to this guy and write something like \psi(t) - \psi(0) = \int_0^t \psi'(s)ds. Can I do this, given only the very bare...
  16. T

    Solving the Schrodinger Equation: WHAT DOES IT ALL MEAN?

    I'm an A-level student (I don't know what the US equivalent is sorry, I'm not an undergraduate is what I'm saying), and I've independently done a project on wave functions for a few simple stationary systems; particle in a box and quantum harmonic oscillator are the ones I focused on in the end...
  17. T

    Bra-ket notation Schrodinger Equation

    This is a pretty trivial question, but how is the Schrodinger equation written out in full, time dependency and all in Dirac notation? I'm interested in this from a purely aesthetic point of view but I'm also a bit confused as to what the bras and the kets really are.
  18. D

    3D Schrodinger Equation of a Wire for Energy Levels - Quantum Physics

    Homework Statement Given a wire with length a and square base b x b (where a >> b), show that the first 1700 (approximately) levels of the electron in the wire are identical for the one dimensional box, when a = 1m and b = 1mm. Homework Equations I know that the allowed energies of...
  19. U

    Solution of Schrodinger equation for free electron

    Hi, i am beginning elementary Quantum Mechanics as my course. While studying one question arise in my mind : In the solution of Schrodinger wave equation there are two parts. ψ=A*exp(jKx) + B*exp(-jkx). (for confined electron) But when dealing with free electron the solution is of...
  20. G

    Schrodinger equation normalization to find A -Griffiths

    Homework Statement In David Griffiths Introduction to Quantum Mechanics (2nd ed.), page 32 he normalizes a time independent wave function to get the coefficient A. He dropped the sine part of the integration with no explanation. What is the justification. Homework Equations The time...
  21. M

    Show that this function satisfies Schrödinger equation for hydogen?

    Homework Statement Show that function Y = C sin θ (5cos2θ - 1 )ei\phi satisfies Scrhödinger equation for hydrogen? Homework Equations The Attempt at a Solution I derivated the required elements of the equation but ended up to some messy equation with lots of sines and...
  22. R

    Time dependant Schrodinger Equation Leapfrog Method

    Hi, I'm trying to set up a programme to compute the numerical solution to the time dependant schrodinger equation of a ground state wave packet in a harmonic oscillator using the leapfrog method. I've been at it for two weeks trying different methods and I'm starting to get extremely stressed...
  23. X

    Schrödinger equation in momentum space and number of solutions

    Hi, Note: I will be sloppy with constant factors in this post. Only the general structure of the equations matters. Consider a particle in a linear potential, \frac{\mathrm d^2}{\mathrm d x^2} \psi(x) + x \psi(x) - E \psi(x) = 0. Mathematically, this is a second-order ODE, and there...
  24. I

    I need just ONE analytic solution to the time-dependent Schrodinger Equation.

    I have been trying to find an analytic solution to the time-dependent Schrodinger Equation. I plan to make a movie of the probability function as it changes over time, but I can't seem to find any analytic solution for the wave function. Is it possible to solve the time-dependent Schrodinger...
  25. A

    Solving Schrodinger Eq.: Quantum #s for n=0,1,2

    Homework Statement The solution of the Schrodinger equation for atom depends on four quantum numbers: the principal n, the orbital l, magnetic m1, and the spin Ms n = 1, 2, 3, 4, ... (integers) l = 0, 1, 2, ... (n-1) m1 = -1, -1+1,...0...1-1, 1 Ms = -1/2, 1/2 List all possible values of...
  26. S

    Newbie Schrodinger equation question

    I discovered physics at a later age and am trying to learn more about Schrodinger's equation, here it is in one dimension -- got to crawl before you walk, I guess -- http://scienceworld.wolfram.com/physics/SchroedingerEquation.html == my question is the variable that is multiplied by i x h bar...
  27. G

    Schrödinger equation for particle on a ring in a magnetic field

    hi i need the schrödinger equation for a particle(electron) in a ring under the influence of a magnetic field that goes through perpendicular to the plane of the ring and i want to consider the spin too. Well, the particle in the ring is pretty easy: - \frac{ \hbar^2}{2mr^2}...
  28. andrewkirk

    Why doesn't wave function collapse contradict Schrodinger equation?

    The postulates of quantum mechanics include: (1) Schrodinger's equation describes how the wave function of a system changes over time, and appears to make the wave function continuous over time. (2) When a measurement is made of quantity m, the wave function instantly changes to an...
  29. S

    Solving 2-D Schrodinger Equation?

    Homework Statement A particle of mass m is confined in a two-dimensions by the potential energy V = 1/2k(x2+4y2). Write down the Schrodinger equation for the system. Write down the ground state wave function and find the lowest four energy levels in terms of the quantities ħ, k, m etc. Make...
  30. L

    Schrodinger Equation: No Higher Powers of ψ

    ψ and its derivatives occur only linearly in the Schrodinger equation, that is, second or higher powers of these quantities do not appear in the equation. Schrodinger equation for a free particle is i\hbar∂ψ(x,t)/∂t = (-\hbar2/2m)(∂2ψ(x,t)/∂x2) Here (∂2ψ(x,t)/∂x2) is second power of ψ. Then...
  31. C

    Schrodinger equation for potential drop

    Homework Statement Hello! I'm looking at a situation where there is a finite potential Vo for x<0, but zero potential for x>0. For a particle moving from left to right, I'm wondering what coefficients for the solution to the Schrodinger equation are equal to zero, and also how to prove that...
  32. J

    Schrodinger Equation: Particle in a Box

    Homework Statement We have been examining a one-dimensional infinite square well where the infinite walls are located at -b and +b. The energy levels in this quantum system are non-degenerate, that is for each energy there is only one wave function. Let us place an infinite potential step...
  33. G

    Energy from schrödinger equation

    hi i was currently thinking about this step here: \langle \psi_n, - \Delta \psi_n \rangle = \int |\nabla \psi_n|^2 dx how do you get from the laplacian to this other expression?
  34. R

    Time-independent Schrodinger Equation

    Why is it that we assume that the solutions to the time-independent Schrodinger Equation are real? Why can't they be complex?
  35. S

    Understanding Time-Independent Schrodinger Equation

    In quantum mechanics, time independent Schrodinger equation gives dynamics of system. How do one claim that this equation is evolution of a system? Since dynamics need time dependency. How do one explain this discrepancy? Thanks in well advance...
  36. S

    Solving the Schrodinger Eq: Time Dependency & System Dynamics

    In quantum mechanics, time independent Schrodinger equation gives dynamics of system. How do one claim that this equation is evolution of a system? Since dynamics need time dependency. How do one explain this discrepancy? Thanks in well advance...
  37. G

    Schrodinger equation significance

    Hi, could someone please explain the equation to me (its significance, application and the how the equation itself works) in simple terms? I find most other resources too complex for me to digest. I hope this is not too much to ask.
  38. N

    Schrodinger Equation for a Bra Vector

    The Schrodinger equation for the complex conjugate of a ket vector is: d/dt<sai(t)| = -(<sai(t)|H)/(i*hcross) How do you derive the above equation from the normal form of the schrodinger equation? I'm mostly confused by where the negative sign is coming from. Thanks
  39. K

    How Do You Solve the Schrodinger Equation with Spherical Coordinates?

    Homework Statement See attached photo The Attempt at a Solution So I have no idea if I have even started this problem correctly so any help would be nice. My working is set out in one of the pictures. Any help would be appreciated I really am not quite sure what to do. Can't...
  40. S

    Non-linear Schrodinger Equation (Differential Equation)

    Hello, i am working on pulse propagation in optical fiber. i have to simulate the nonlinear Schrodinger equation using the FDTD (Finite Difference Time Domain) method. The Schrodinger equation has the form dA/dz = i/2 β2 d2A/dt2 –α/2 +iγ |A2|A where β2 is dispersion, α is...
  41. K

    How Do You Solve the Schrödinger Equation Using Polar Coordinates?

    Homework Statement See attached photo Homework Equations none The Attempt at a Solution Just hoping someone could give me a hand with this. I think I need to convert polar coords into y and z using trig then perform an addition of the wavefunctions as the linear transform...
  42. N

    Two Particle Schrodinger Equation in Polar Coordinates

    Homework Statement Consider two non-identical, non-interacting particles of mass M that are constrained to move on a circle of radius R. Write down the Schrodinger equation for this problem and find the eigenfunctions and energy levels of this system. Homework Equations (see below)...
  43. S

    Range of validity of the Schrodinger equation

    I understand that the Schrodinger equation describes the behaviour of matter and the electromagnetic wave down to the microscopic scale. But I'm not sure about the everyday sound waves, water waves, wave on a spring, etc. What do you think?
  44. R

    Schrodinger equation and atoms

    To what extent does the Schrodinger equation apply to an atom as a whole?
  45. A

    Time invariance of Schrodinger equation

    The Schrodinger equation is linear in time. I was wondering if that means that is not invariant under time reversal. That would be a surprise because all other microscopic laws (Maxwell's equations, Newton's equations) are time invariant. Can you please clear this doubt?
  46. N

    Solving schrodinger equation for quarkonium

    I've tried to solve Schrodinger equation for Charmonium and Bottomonium and there are some problems with that : 1. As we know the Schrodinger equation is independent from quantum number "ml" so there would be same Energy for different "ml" for a specific n. But what we see in PDG book is...
  47. G

    Time derivative of schrodinger equation

    Why is the TDSE first derivative in time. Now I know that it is required so that the wave functions are complex... but is there any physical interpretation for this requirment??
  48. C

    3D Schrödinger Equation for a magnetic vector potential in cylinder coordinates

    Hi there, during my work on my PhD thesis as an experimental physicist I ended up with a very theoretical problem: What does the wavefunction of an electron traveling through a magnetic vector potential look like? I chose a cylindrical coordinate system with a magnetic vector potential A...
  49. 0

    Schrodinger Equation in momentum space? ?

    Schrodinger Equation in momentum space? ?? I don't know if this makes any sense at all, but I'm studying QM and just trying to generalize some things I'm learning. Please let me know where I go wrong.. Basically by my understanding the most general form of the Schrodinger Equation can be...
  50. A

    Matching conditions for solutions to the Schrodinger equation

    In standard, run-of-the-mill, one-dimensional scattering problems (e.g., finite square wells), we calculate transmission and reflection amplitudes by (in part) making sure that our wave function \psi satisfies the following conditions at discontinuities of the potential: (1) It is continuous...
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