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HW9. Show that the time-independent Schrödinger equation is given by P(x)/(x) =...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
4. Consider the time-independent Schrödinger equation for an "atom" in which the attractive force...
4. Consider the time-independent Schrödinger equation for an "atom" in which the attractive force between the electron and the proton is modeled as a spring. Then V(r)- (1/2)mu22, where m is the mass of the electron and w is the natural frequency of oscillation You're goal is to determine the eigenenergies of the electron and the corresponding wave functions, as outlined below Let's again start with the radial equation associated with the Schrodinge equation 4.37 in Griffiths [where u(r) R(r)...
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation,...
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation, which in position space reads iħ 4(x, t) = -24(x, t) + V(x, t)(x, t). ih vrt - h ? a) Consider any two normalized solutions to the Schrödinger equation, 41(2, t) and 02(3,t). Prove that their inner product is independent of time, doo 1 Vi (2, t)u2(x, t) dc = 0. dt J-00 Hint: prove the useful intermediate result, a 202 201 -...
Potential energy function, V(x) = (1/2)mw2x2 Assuming the time-independent Schrödinger equation, show that the following wave...
Potential energy function,
V(x) = (1/2)mw2x2
Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the particle, E ita energy, (z) the potential (a) Consider a particle moving in a constant pote E> Vo, show that the following wave function is a solution of the TISE and determine the relationahip betwoen E an zero inside the well, ie. V(2)a 0foros L, and is infinite ou , ie, V(x)-w (4) Assuming (b) Consider an infinite square well with walls at 1-0...
You are given the equation used to solve a problem: (9.0x10° N mº/c?)(2.0x10-C)(1.60x10-19 C) 1(1.67 x...
You are given the equation used to solve a problem: (9.0x10° N mº/c?)(2.0x10-C)(1.60x10-19 C) 1(1.67 x 10-27 kg)(4.0 x 10 m/s)? +0= }(1.67 x 10-27 kg)v? +! 0.0010 m Part A Choose a realistic problem for which this is the correct equation. m/s. What is the protons A proton is shot toward a 2.0 nC point charge with a speed of 4.0 x 10 speed when it is 1.0 mm from the charge? m/s. What is the A hydrogen atom...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
4. Consider the time-independent Schrödinger equation for an "atom" in which the attractive force between the electron and the proton is modeled as a spring. Then V(r)- (1/2)mu22, where m is the mass of the electron and w is the natural frequency of oscillation You're goal is to determine the eigenenergies of the electron and the corresponding wave functions, as outlined below Let's again start with the radial equation associated with the Schrodinge equation 4.37 in Griffiths [where u(r) R(r)...
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation, which in position space reads iħ 4(x, t) = -24(x, t) + V(x, t)(x, t). ih vrt - h ? a) Consider any two normalized solutions to the Schrödinger equation, 41(2, t) and 02(3,t). Prove that their inner product is independent of time, doo 1 Vi (2, t)u2(x, t) dc = 0. dt J-00 Hint: prove the useful intermediate result, a 202 201 -...
Potential energy function,
V(x) = (1/2)mw2x2
Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the particle, E ita energy, (z) the potential (a) Consider a particle moving in a constant pote E> Vo, show that the following wave function is a solution of the TISE and determine the relationahip betwoen E an zero inside the well, ie. V(2)a 0foros L, and is infinite ou , ie, V(x)-w (4) Assuming (b) Consider an infinite square well with walls at 1-0...
You are given the equation used to solve a problem: (9.0x10° N mº/c?)(2.0x10-C)(1.60x10-19 C) 1(1.67 x 10-27 kg)(4.0 x 10 m/s)? +0= }(1.67 x 10-27 kg)v? +! 0.0010 m Part A Choose a realistic problem for which this is the correct equation. m/s. What is the protons A proton is shot toward a 2.0 nC point charge with a speed of 4.0 x 10 speed when it is 1.0 mm from the charge? m/s. What is the A hydrogen atom...
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