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Particle in a Box Class Objectives Introduce the idea of a free particle. Solve the TISE for the free particle case. Particle in a Box The best way to understand Schrödinger’s equation is to solve it for various potentials. Particle in a Box The best way to understand Schrödinger’s equation is to solve it for various potentials. The simplest of these involving forces is particle confinement (a particle in a box). Particle in a Box Consider a particle confined along the x axis between the points x = 0 and x = L. Particle in a Box Consider a particle confined along the x axis between the points x = 0 and x = L. Inside the box it free but at the edges it experiences strong forces which keep it confined. Particle in a Box Consider a particle confined along the x axis between the points x = 0 and x = L. Inside the box it free but at the edges it experiences strong forces which keep it confined. Eg. A ball bouncing between 2 impenetrable walls. Particle in a Box Consider a particle confined along the x axis between the points x = 0 and x = L. Inside the box it free but at the edges it experiences strong forces which keep it confined. Eg. A ball bouncing between 2 impenetrable walls. U E 0 L Particle in a Box U E 0 L E: total energy of the particle U: potential containing the particle. The Pd of the walls. E<U Particle in a Box U E 0 L E: total energy of the particle U: potential containing the particle. The Pd of the walls. E<U Inside the well the particle is free. Particle in a Box U E 0 L E: total energy of the particle U: potential containing the particle. The Pd of the walls. E<U Inside the well the particle is “free”. This is because U (x) is zero inside the well. Particle in a Box U E 0 L E: total energy of the particle U: potential containing the particle. The Pd of the walls. E<U Inside the well the particle is “free”. This is because U (x) is zero inside the well. IncreasingU (x) to infinity as the width is reduced to zero, we have the idealization of an infinite potential square well. Infinite square potential 0 L U x Particle in a Box Classically there is no restriction on the energy or momentum of the particle. Particle in a Box Classically there is no restriction on the energy or momentum of the particle. However from QM we have energy quantization. Particle in a Box We are interested in the time independent waveform (x) of the particle. The particle can never be found outside the well. Ie. ( x) 0 in the region 0 x L Particle in a Box • Since U ( x) 0 we get that, d 2 E ( x) 2 2m dx we take, 2mE k 2 2 Particle in a Box • • Since U ( x) 0 we get that, d 2 E ( x) 2 2m dx d 2 So that dx 2 we take, k 2 ( x) 2mE k 2 2 Particle in a Box The solutions to this equation are of the form ( x) A sin kx B cos kx for 0 x L (a linear combination of cosine and sine waves of wave number k) Particle in a Box The solutions to this equation are of the form ( x) A sin kx B cos kx for 0 x L (a linear combination of cosine and sine waves of wave number k) The interior wave must match the exterior wave at the boundaries of the well. Ie. To be continuous! Particle in a Box Therefore the wave (x) must be zero at the boundaries, x=0 and x=L. Particle in a Box Therefore the wave (x) must be zero at the boundaries, x=0 and x=L. At x=0, ( x) A sin 0 B cos 0 0 Particle in a Box Therefore the wave (x) must be zero at the boundaries, x=0 and x=L. At x=0, ( x) A sin 0 B cos 0 0 Particle in a Box Therefore the wave (x) must be zero at the boundaries, x=0 and x=L. At x=0, ( x) A sin 0 B cos 0 0 B 0 Particle in a Box Therefore the wave (x) must be zero at the boundaries, x=0 and x=L. At x=0, ( x) A sin 0 B cos 0 0 B 0 At x=L, ( x) A sin kL 0 Particle in a Box Therefore the wave (x) must be zero at the boundaries, x=0 and x=L. At x=0, ( x) A sin 0 B cos 0 0 B 0 At x=L, ( x) A sin kL 0 Since A 0 , then sin kL 0 kL n, n 1,2,... 2E 2 Recall: k 2 Particle in a Box From this we find that particle energy is quantized. The restricted values are 2 k 2 n 2 2 2 En 2m 2mL2 Particle in a Box From this we find that particle energy is quantized. The restricted values are 2 k 2 n 2 2 2 En 2m 2mL2 Note E=0 is not allowed! Particle in a Box From this we find that particle energy is quantized. The restricted values are 2 k 2 n 2 2 2 En 2m 2mL2 Note E=0 is not allowed! N=1 is ground state and n=2,3… excited states. Particle in a Box Finally, given k and B we write the waveform as nx n ( x) A sin L Particle in a Box Finally, given k and B we write the waveform as nx n ( x) A sin L We need to determine A. Particle in a Box Finally, given k and B we write the waveform as nx n ( x) A sin L We need to determine A. To do this we need to normalise. Particle in a Box Normalising, L * n ( x) n ( x)dx 1 0 Particle in a Box Normalising, L * n ( x) n ( x)dx 1 0 nx A sin dx 1 L 0 L 2 2 Particle in a Box Normalising, L * n ( x) n ( x)dx 1 0 nx A sin dx 1 L 0 2 L A 1 2 L 2 2 Particle in a Box Normalising, L * n ( x) n ( x)dx 1 0 nx A sin dx 1 L 0 2 L 2 L A 1 A 1 2 2 L 2 2 Particle in a Box Normalising, L * n ( x) n ( x)dx 1 0 nx A sin dx 1 L 0 2 L 2 L A 1 A 1 2 2 L 2 2 Particle in a Box Normalising, L * n ( x) n ( x)dx 1 0 nx A sin dx 1 L 0 2 2 L A 1 A L 2 2 nx n sin L L L 2 2 Particle in a Box For each value of the quantum number n there is a specific waveform (x ) describing the state of a particle with energy En . Particle in a Box For each value of the quantum number n there is a specific waveform (x ) describing the state of a particle with energy En . The following are plots of nvs x and 2 the probability density n vs x. Particle in a Box