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Electron in an Infinite Potential Wells
1. Start with an infinite well. (Setup: Infinite Well). Go to View. Choose energy, position, values/dimensions, state phasors, expectation values, uncertainties, wavefunction – probabilities+phase. The top panel shows the energy level diagram for the well, i.e. the allowed energies for a particle trapped in the well. The middle panel shows the wave function of the particle in the well. On the main screen choose mouse = set eigenstate and Ground State. Make sure you understand why the colour of the graph changes with time and what the phase indicator at the bottom of the page is indicating [Hint: consider Eq. 1. You could also temporarily change the View>Wavefunction>Real+imaginary]
2. Solving the time-independent form of Schrödinger’s equation for a particle in an infinite well of width L leads to the result that the lowest energy the particle can have (the energy of the ground state n=1) is inversely proportional to L2. Use the program to verify this result by measuring the energy for at least three L values. Record well width, particle mass, and n. [Hint: maximize window, and drag the top of middle panel down to expand energy panel]
3. Solving the time-independent form of Schrödinger’s equation for a particle in an infinite well of width L leads to the result that the energy of the particle is quantised. If the positive integer n labels the stationary quantum states of the particle then the energy is proportional to n2. Use the program to verify this result, recording the fixed parameters. [Hint: stationary states can be set by clicking on the energy level, or by double-clicking the appropriate phasor circle at the bottom]
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4. An electron is in a well of width 1.6 nm and is in the state n = 1. The electron can gain energy by absorbing a quantum of electromagnetic radiation. What frequency of light is required to cause the electron in state n = 1 to change to being in the state n = 2? Relate this frequency to the frequencies of the wavefunctions for states 1 and 2.
5. When the cursor is moved over the energy level diagram the wavefunction associated with a particular energy level is displayed as a yellow curve. The wavefunction takes positive and negative values and has units of 1/nm1/2. It equals 0 at the edges of the potential well and is 0 outside the well. How is the amplitude of the wavefunction (the yellow curve) related to the probability curve for a stationary state? Draw free-hand sketches of the amplitude curves and probability curves for states n = 1 and n = 2.
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