Introduction to Schrödinger: Solving the Time-Dependent Schrödinger Equation

1. Lecture 14An introduction to SchrödingerSolving the time dependent Schrödinger equation • Only 7 lectures left • Read through your notes in conjunction with lecture presentations • Try some examples from tutorial questions or Phils Problems • Email me or come to E47 if anything is unclear Remember Phils Problems and your notes = everything http://www.hep.shef.ac.uk/Phil/PHY226.htm

2. SUMMARY of the procedure used to solve PDEs 1. We have an equation with supplied boundary conditions 2. We look for a solution of the form 3. We find that the variables ‘separate’ 4. We use the boundary conditions to deduce the polarity of N. e.g. 5. We use the boundary conditions further to find allowed values of k and hence X(x). so 6. We find the corresponding solution of the equation for T(t). 7. We hence write down the special solutions. • By the principle of superposition, the general solution is the sum of all special • solutions.. www.falstad.com/mathphysics.html 9. The Fourier series can be used to find the particular solution at all times.

3. Introduction to Quantum Mechanics Something weird is going on……justification of Quantum Mechanics If you were to shine white light through sodium gas you would see an absorption line in the total spectrum. Passing a current through sodium gas causes the emission of monochromatic light Sodium street light Spectral lines come from the fact that atoms absorb or emit particular energies of photons.

4. Introduction to Quantum Mechanics Something weird is going on……justification of Quantum Mechanics Hydrogen gas for example can therefore only absorb or emit specific wavelengths of light. When a photon is absorbed an electron is promoted to a higher energy level, but only specific energy photons can do this, it follows that discrete energy levels exist in the atom. This provides us with a spectral fingerprint that tells us about the inner structure of the atom. E=hf Looking at the continuous spectrum and comparing it with the Hydrogen emission spectrum we see that the five emission lines are at 1.9, 2.5, 2.8, 3.0, 3.1eV.

5. Introduction to Quantum Mechanics Something weird is going on……justification of Quantum Mechanics The energy levels are said to be quantised and when the electron moves from one energy level to another they are said to make quantum jumps. For example when a current is passed through sodium gas, electrons are promoted to higher energy levels, they then drop to lower energy levels and emit discrete energies or wavelengths of light. The energy of the photons emitted are equal to the differences between the energy levels so the 2.1eV photons emitted by street lights tell us that there is an energy gap between levels of 2.1eV.

6. Introduction to Quantum Mechanics Something weird is going on……justification of Quantum Mechanics The usual position of the electron in the hydrogen atom is in the ground state E1 whereas E∞ corresponds to the situation where the electron is separated from the proton by an infinite amount. five emission lines are at 1.9, 2.5, 2.8, 3.0, 3.1eV NB. The energy of the emission line is determined by ΔE (the difference in energy states) Although there are 5 visible lines there are many more in the EM spectrum E∞ - E1 corresponds to the ionisation energy which is 13.6eV for Hydrogen. Once the electron has been ionised, any remaining energy increases the kinetic energy of the proton and electron which, existing in the continuum region can have any value of energy.

7. Introduction to Quantum Mechanics Prediction of atomic structure Lots of people tried to explain atomic structure before quantum mechanics came along. Rutherford’s atom had a central positive nucleus with a negative electron orbiting it. The electrical attraction held them together and the rotational motion kept them apart. In the model the electron can exist at any radius so long as the rotational velocity increases to compensate. This model therefore does not predict discrete energy levels or explain line spectra.

8. Introduction to Quantum Mechanics Prediction of atomic structure Bohr then developed the Newtonian and electromagnetic ideas of electron motion and managed to construct a Hydrogen atom with the discrete energy levels required for line emission. The model combines classical orbiting electron with quantized electron momenta. The electrons therefore still orbit the nucleus but in this model they are restricted to specific radii. However when an electron rotates it constantly undergoes acceleration and emits electromagnetic radiation. Therefore we would expect the electron to constantly emit photons, losing energy constantly and eventually spiralling into the nucleus. This doesn’t happen!!

9. Introduction to Quantum Mechanics Quantum indeterminacy If you bombard a hydrogen atom with photons of high enough energy to promote the electron from E1 to E3 then sometimes it will do this and other times it wont !!! The same occurs for an electron in an excited state that can either drop down one or more energy levels. We can never know if an individual atom has absorbed a photon or not and the best we can do based on statistics is to assign a probability to whether or not the process will occur.

10. Introduction to Quantum Mechanics Schrödinger equation Both the Rutherford and Bohr models of the atom are therefore flawed. In the 1920s a group of Physicists headed by Schrodinger developed what we now know as the Schrodinger equation. The equation did two main things. It predicted the energy levels of the H atom. But it also introduced the concept that the behaviour of the electron is intrinsically indeterminate. According to Quantum mechanics, if a H atom has a certain amount of energy it is impossible to say in advance of the measurement what value will be obtained for the electrons position and momentum. This means that if we perform identical measurements on an atom with the same energy, we will always have different outcomes. What can be predicted are the range of possible outcomes of the measurement and the probability of each of these outcomes.

11. Introduction to Quantum Mechanics Schrödinger equation A good way of illustrating the uncertainty of the position of the electron is to show it in 2D as an electron cloud. If we imagine that for a single atom we measure the position of the electron 1000 times and place a single dot on the paper at each location we found the electron to be then we would end up with an electron probability cloud. The greater the concentration of dots, the more likely the electron would be found at this location. The maximum concentration of dots corresponds to a radius of 5.29×10-11m which is exactly the radius predicted by the Bohr model when the electron is in its ground state.

12. Introduction to Quantum Mechanics Schrödinger equation So far we’ve only talked about the ground state but the Schrodinger equation can be used to predict the behaviour of the electron in any of its energy states (eigenvalues). The figure right shows the probability density clouds of the ground state of the H atom and also other higher energy states. But is this really the best way we can represent the likelihood that an electron will be in a certain position at a certain time? The use of electron probability clouds to predict the probability associated with measuring in the quantum world is visually very clear but nowhere near as useful as the eigenfunctions we are now so familiar with.

13. Introduction to Quantum Mechanics The electron probability cloud is analogous to the probability density function given below, expressing the probability P of finding the particle at some particular location between b and a. Remembering that The position of the particle is, of course, one quantity we might imagine measuring experimentally. It is an observable quantity. But there are many physical observables. One is the energy, which we determine from the solutions of the TISE. We usually see it written like this:

14. OK, so I know that I have to start thinking about operators and eigenfunctions and eigenvalues but where do we start ????!! What you have to realise is that because there is a probability associated with pinning the particle down to a specific energy or position or momentum, with every calculation we need to incorporate the probability associated with the measurement. We do this using OPERATORS such as the one below….. You are most familiar with using the TISE to find the specific energy levels associated within a zero potential well where: where H denotes the Hamiltonian operator The whole point of Quantum mechanics !!!!!!!!! Quantum mechanics does not explain how a quantum particle behaves. Instead, it gives a recipe for determining the probability of the measurement of the value of a physical variable (e.g. energy, position or momentum). This information enables us to calculate the average value of the measurement of the physical variable.

15. Introduction to Quantum Mechanics Operators play a crucial role in the theory of quantum mechanics, as each experimental observable is associated with an operator. A “hat” usually denotes operators. An equation of the form is called an eigenvalue equation. The values of are called the eigenvalues, and the corresponding functions are called the eigenfunctions. The allowed values of the observable are the eigenvalues of the operator, each corresponding to a function (the eigenfunction) which represents the state of the system when the observable has that value. The TISE is such an equation. The allowed values of energy are the eigenvalues of the Hamiltonian operator, and the corresponding wavefunctions are its eigenfunctions. Eigenvalue Eigenvalue equation Eigenfunction

16. Introduction to Quantum Mechanics Let’s show how we can find the eigenvalues of energy in zero V using an operator …. Operator Eigenvalue equation Eigenfunction So as expected Eigenvalue

17. Introduction to Quantum Mechanics The list of operators is given below: If we measure the momentum of the momentum eigenfunction, The operator is, and so Here p is the eigenvalue, and ψ(x) is the eigenfunction so

18. Introduction to Quantum Mechanics Infinite potential well The particle cannot exist where the potential is infinite, so the boundary conditions are: and write Re-arrange as where As always set so and so Boundary conditions are General solution is

19. Introduction to Quantum Mechanics Thus the solutions are Infinite potential well The energies are given by where as eigenfunctions eigenvalues Notice how as a consequence of the boundary conditions on ψ(x) at x = 0 and L we must fit an integral number of half-wavelengths into the potential well of width L

20. Introduction to Quantum Mechanics Infinite potential well For each eigenfunction the probability of finding the particle in the well is unity. Thus, This determines A: so so so and therefore So the eigenfunctions and their corresponding eigenvalues have been found using the kinetic energy operator

21. Introduction to Quantum Mechanics Notice that E = KE + PE, thus KE(x) = E - PE = E -V(x) Finite potential well Eigenfunctions with energy eigenvalues E > V0 are unbound Eigenfunctions with energy eigenvalues E < V0 are bound. For bound states the wavefunction penetrates the classically forbidden region. Thus, the particle exists in a region where its kinetic energy is negative. To find energies of these states we’ll solve the time independent Schrödinger equation:

22. Introduction to Quantum Mechanics Finite potential well REGION I We need to solve: In region I solutions are IPW Using same technique as for infinite well but for different boundary conditions NB. The value k above is actually smaller than the corresponding value k for the inifinite potential well. This means that is not 0 at the potential well boundaries. Also since the energy levels are lower compared to IPW FPW

23. Introduction to Quantum Mechanics Finite potential well REGION II We need to solve: Re-arrange to where As always set so and so The general solution is then

24. Introduction to Quantum Mechanics Finite potential well REGION II Boundary conditions to satisfy max probability of 1, thus C = 0. must be continuous at boundary between regions I and II at x = L/2. (i) so (ii) so Dividing eqn (ii) by eqn (i) we eliminate A and D to obtain the condition

25. Introduction to Quantum Mechanics Finite potential well summary REGION I REGION II If we wanted to we could then perform the same procedure for the region to the left of the potential well. We would then be able to normalise the function between ± ∞ to unity and find another expression linking the coefficients A and D 1/α is defined as the penetration depth

26. Introduction to Quantum Mechanics Potential wells

27. Introduction to Quantum Mechanics Quantum Tunnelling Classically, if you have a potential barrier of height V and a particle incident on that barrier with E < V, the particle would reflect off the barrier completely. The same system in quantum mechanics gives a non-zero probability that the particle will be transmitted through the barrier. This is a wave phenomenon, but in quantum mechanics particles exhibit wave-like properties. The wavefunction of the tunneling particle decreases exponentially in the barrier. The tunneling probability is strongly dependent on the width of the barrier, the mass of the particle, and the quantity (V-E). For instance, the ratio of tunneling probability for protons to electrons is around a factor of 10-91.

28. Introduction to Quantum Mechanics Quantum Tunnelling: uses The most important applications of quantum tunnelling are in semiconductor and superconductor physics. Phenomena such as field emission, important to flash memory, are explained by quantum tunnelling. Another major application is in scanning tunnelling microscopes which can resolve objects that are too small to see using conventional microscopes, overcoming the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunnelling electrons.

29. Introduction to PDEs In many physical situations we encounter quantities which depend on two or more variables, for example the displacement of a string varies with space and time: y(x, t). Handing such functions mathematically involves partial differentiation and partial differential equations (PDEs). Wave equation Elastic waves, sound waves, electromagnetic waves, etc. Schrödinger’s equation Quantum mechanics Diffusion equation Heat flow, chemical diffusion, etc. Laplace’s equation Electromagnetism, gravitation, hydrodynamics, heat flow. Poisson’s equation As (4) in regions containing mass, charge, sources of heat, etc.

30. SUMMARY of the procedure used to solve PDEs 1. We have an equation with supplied boundary conditions 2. We look for a solution of the form 3. We find that the variables ‘separate’ 4. We use the boundary conditions to deduce the polarity of N. e.g. 5. We use the boundary conditions further to find allowed values of k and hence X(x). so 6. We find the corresponding solution of the equation for T(t). 7. We hence write down the special solutions. • By the principle of superposition, the general solution is the sum of all special • solutions.. www.falstad.com/mathphysics.html 9. The Fourier series can be used to find the particular solution at all times.

31. Solving the time dependent Schrödinger equation The TDSE is a linear equation, so any superposition of solutions is also a solution. For example, consider two different energy eigenvalues, with energies E1 and E2. Their complete normalised wavefunctions at t = 0 are: But any superposition such as also satisfies the TDSE, and thus represents a possible state of the system. Recall that all wavefunctions must obey the normalization condition: When we superpose, the resulting wavefunction is no longer normalised. However it can be shown that the normalisation condition is fulfilled so long as:

32. Solving the time dependent Schrödinger equation Consider the time dependent Schrödinger equation in 1 dimensional space: Within a quantum well in a region of zero potential, V(x,t) = 0, this simplifies to: Question Let’s solve the TDSE subject to boundary conditions Y(0, t) = Y(L, t) = 0 (as for the infinite potential well) For all real values of time t and for the condition that the particle exists in a superposition of eigenstates given below at t = 0 .

33. Solving the time dependent Schrödinger equation n = 1 Question Let’s solve the TDSE subject to boundary conditions Y(0, t) = Y(L, t) = 0 (as for the infinite potential well) and for the condition that the particle exists in a superposition of eigenstates given below at t = 0 . n = 2 Superposition at t = 0 n = 3

34. Solving the time dependent Schrödinger equation In a region of zero potential, V(x,t) = 0, so : Step 1: Separation of the Variables Our boundary conditions are true at special values of x, for all values of time, so we look for solutions of the form Y(x, t) = X(x)T(t). Substitute this into the Schrödinger equation: Step 2: Rearrange the equation Separating variables:

35. Solving the time dependent Schrödinger equation Step 3: Equate to a constant Now we have separated the variables. The above equation can only be true for all x, t if both sides are equal to a constant. It is conventional (see PHY202!) to call the constant E. So we have which rearranges to (i) (ii) which rearranges to Step 4: Decide based on situation if E is positive or negative We have ordinary differential equations for X(x) and T(t) which we can solve but the polarity of N affects the solution ….. For X(x) Our boundary conditions are Y(0, t) = Y(L, t) = 0, which means X(0) = X(L) = 0. So clearly we need E > 0, so that equation (i) has the form of the harmonic oscillator equation. It is simpler to write (i) as where giving

36. Solving the time dependent Schrödinger equation Step 5: Solve for the boundary conditions for X(x) For X(x) Our boundary conditions are Y(0, t) = Y(L, t) = 0, which means X(0) = X(L) = 0. If where then applying boundary conditions gives X(0) = 0 gives A = 0 ; we must have B ≠ 0 so X(L) = 0 requires , i.e. so for n = 1, 2, 3, …. Step 6: Solve for the boundary conditions for T(t) Equation (ii) has solution as it’s only a 1st order ODE Step 7: Write down the special solution for Y (x, t) where (These are the energy eigenstates of the system.)

37. Solving the time dependent Schrödinger equation Step 8: Constructing the general solution for Y (x, t) We have special solutions: The general solution of our equation is the sum of all special solutions: Superposition at t = 0 (In general therefore a particle will be in a superposition of eigenstates.) Step 10: Finding the particular solution for all times If we know the state of the system at t = 0, we can find the state at any later time. Since we said that Then we can say where and

38. Particular solution to the time dependent Schrödinger equation Y3 Y2 1st Eigenfunction Y1 3rd Eigenfunction 2nd Eigenvalue 3rd Eigenvalue 1st Eigenvalue