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Particle in a Box - 1D (14.3)

Particle in a Box - 1D (14.3). A simple problem that demonstrates the principles of quantum mechanics is the particle in a box (PIB) Inside the box of a given length ( a ), the potential is zero; outside the box the potential is infinite

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Particle in a Box - 1D (14.3)

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  1. Particle in a Box - 1D (14.3) • A simple problem that demonstrates the principles of quantum mechanics is the particle in a box (PIB) • Inside the box of a given length (a), the potential is zero; outside the box the potential is infinite • The wavefunction must be zero outside the box since the PE can’t be infinite • The solution to the SE for the PIB is an oscillating function • It is sinusoidal since the wavefunction must be zero at the ends of the box • An arbitrary integer (n) is in the solution and is referred to as a quantum number • The energy of the PIB is also dependent on the quantum number • Quantum number means the energy is quantized (only has certain values) • Energy also depends on the length of the box

  2. Properties of PIB Wavefunctions (14.3-14.4) • Each solution of the PIB SE represents a state of the system • The lower the state (smaller n) the lower the energy of the system • First state is referred to as the ground state; higher energy states are called excited states • Some interesting features arise from the solution of the PIB SE • As the quantum number increases, so does the number of values of x for which the PIB wavefunction equals zero (nodes) • The PIB probability densities show a number of interesting characteristics (non-uniform probabilities, regions of zero probability within the box) • PIB for higher dimensions (2D, 3D) show very similar behavior • Wavefunction is a product of 1D PIB wavefunctions • Energy is a sum of 1D PIB energies • Some energy levels are degenerate (have same energies)

  3. Excited States and Spectroscopy (18.1-18.2) • In order for a system to go from one state to another, it must absorb or emit a quantized amount of energy (radiation) • Light is often used to promote particles to higher states (absorption) • Light is often emitted when excited states go back to lower lying states (emission) • Spectroscopy is the use of light to probe states in matter • Type of light needed depends on the physical problem one is interested in (i.e., the potential energy function) • Selection rules dictate whether two states can be connected by shining light on it (depends on the nature of the wavefunctions of the two states) • Does PIB model any real physical phenomena? • The spectroscopy of highly conjugated molecules (e.g., polyenes) can be explained quite well by PIB model

  4. Particle in a Box

  5. PIB Wavefunctions

  6. PIB Probability Densities

  7. Absorption and Emission of Radiation

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