1. Modern Physics lecture 4

2. The Schroedinger Equation As particles are described by a wave function, we need a wave equation for matter waves As particles are described by a wave function, we need a wave equation for matter waves The general form for a wave travelling in one dimension is The general form for a wave travelling in one dimension is

3. The Schroedinger Equation For a matter wave moving on the x-axis we can write the full wave function as For a matter wave moving on the x-axis we can write the full wave function as For our purpose we will look only at the spatial part of this wave function which satisfies the wave equation For our purpose we will look only at the spatial part of this wave function which satisfies the wave equation This is the Schrodinger Equation (E is kinetic and U is potential energy) This is the Schrodinger Equation (E is kinetic and U is potential energy)

4. A particle in a box 1. Classically

5. A particle in a box Classically if we put a particle in a box, and it is confined to move in one direction. The particle will reflect off the walls and rattle backward and forward in the box Classically if we put a particle in a box, and it is confined to move in one direction. The particle will reflect off the walls and rattle backward and forward in the box Any time that we open the box we find that the particle has equal probability of being somewhere in the box Any time that we open the box we find that the particle has equal probability of being somewhere in the box

6. String Harmonics Recall the harmonics of a string held at both ends Recall the harmonics of a string held at both ends Waves exist only when the wavelength is an integer multiple of half wavelengths Waves exist only when the wavelength is an integer multiple of half wavelengths The wavelength of the standing wave of a string is quantised The wavelength of the standing wave of a string is quantised

7. A particle in a box The wave function for a standing wave is given by The wave function for a standing wave is given by Substituting the quantised formula for the wavelength we arrive at Substituting the quantised formula for the wavelength we arrive at

8. A Particle in a Box The box is constructed from walls that are infinitely high potential energy barriers and in between the potential energy is zero The box is constructed from walls that are infinitely high potential energy barriers and in between the potential energy is zero We can re-write the S.E. for the region between the walls to be We can re-write the S.E. for the region between the walls to be This has a solution This has a solution  Which is the same solution as was found by comparing matter waves to waves in a string

9. A Particle in a Box 2. Quantum mechanics Potential energy: Schroedinger equation inside the box:

10. A particle in a box Quantum mechanically we find the same solution for particles moving inside a box Quantum mechanically we find the same solution for particles moving inside a box The wave function is described by The wave function is described by The probability of finding a particle in a position inside the box when it is opened will therefore be given by The probability of finding a particle in a position inside the box when it is opened will therefore be given by

11. A particle in a box: solution A standing wave

12. A particle in a box Because the deBroglie wavelength is quantised the momentum is also quantised Because the deBroglie wavelength is quantised the momentum is also quantised Therefore kinetic energy is also quantised Therefore kinetic energy is also quantised

13. A particle in a box: solution n = 1, 2, 3,... with The lowest allowed energy corresponds to n=1 therefore it is not zero The lowest allowed energy corresponds to n=1 therefore it is not zero The quantum mechanical lowest energy state is called zero point energy The quantum mechanical lowest energy state is called zero point energy

14. Quantum Mechanics Energy levels are important because if the particle is charged (such as an electron) a photon will be emitted as the particle changes energy levels Energy levels are important because if the particle is charged (such as an electron) a photon will be emitted as the particle changes energy levels It can also absorb a photon raising the energy level It can also absorb a photon raising the energy level This gives us the opportunity of probing quantum effects using spectroscopy This gives us the opportunity of probing quantum effects using spectroscopy

15. A particle in a finite well If the walls of the box are finite If the walls of the box are finite If the energy of the particle is lower than the potential then classically the particle is bound (stuck) in the box (region II) If the energy of the particle is lower than the potential then classically the particle is bound (stuck) in the box (region II) In quantum mechanics sometimes the particle can be found in regions I and III In quantum mechanics sometimes the particle can be found in regions I and III U L 0 x IIIIII E

16. A particle in a finite well In regions I and III the Schrodinger equation is In regions I and III the Schrodinger equation is This has the solution This has the solution

17. A particle in a finite well Thus we arrive at the following interpretation of the solutions Thus we arrive at the following interpretation of the solutions

18. Harmonic oscillator Potential energy SE : solution

19. Harmonic oscillator