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#1: Find the time depended wave functions V(x, t) = ?

#1: Find the time depended wave functions V(x, t) = ?

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The system has a particle (its mass is m) in the vacuum, no potential energy (V(x) = 0). The Hamiltonian of the system ħ² 8² is given by Ĥ = -2m dx² The initial wave functions at time t = 0 is where a is a real constant. Question #1: Find the time depended wave functions V(x, t) = ? Question #2: Also, for the wave function (x, t) (that has been found above), find the probability current density defined as Note: +∞ 1000 +∞ [too where ã and ß are complex constants; v (z,t = 0) = (=) *exp(- 02²) 1 j = 2 2m 8 (¥* (x)pV(x) + V(x) (pV(x))*) r+∞ 88 da exp(-ã+²) = √ dx dr exp (-ãx² - ikx) = = √= exp(-1/²) dk exp (-²+ ika) = √³ exp(-82²) 4 ax²+bx+c = a (x + = a (x + 2)² + c - 50 2a 4a c
Transcribed Image Text:The system has a particle (its mass is m) in the vacuum, no potential energy (V(x) = 0). The Hamiltonian of the system ħ² 8² is given by Ĥ = -2m dx² The initial wave functions at time t = 0 is where a is a real constant. Question #1: Find the time depended wave functions V(x, t) = ? Question #2: Also, for the wave function (x, t) (that has been found above), find the probability current density defined as Note: +∞ 1000 +∞ [too where ã and ß are complex constants; v (z,t = 0) = (=) *exp(- 02²) 1 j = 2 2m 8 (¥* (x)pV(x) + V(x) (pV(x))*) r+∞ 88 da exp(-ã+²) = √ dx dr exp (-ãx² - ikx) = = √= exp(-1/²) dk exp (-²+ ika) = √³ exp(-82²) 4 ax²+bx+c = a (x + = a (x + 2)² + c - 50 2a 4a c
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