The time-independent Schroedinger equation is given by:
− Wave functions that satisfy this equation are called energy eigenstates. a) If U=0 for all positions, this represents a free particle. For a wave function with definite momentum ℏ,, compute E. b) Is the relationship derived from a) consistent with what we know from classical mechanics for a free particle? Explain how or how not. c) Consider the wave function ((^b[j + ^bâj), with A some number and c, d not equal in magnitude or sign. Show whether this wave function is an energy eigenstate. d) What would be the result of a measurement of the momentum of the wave function in part c)? The energy?
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The time-independent Schroedinger equation is given by:
− Wave functions that satisfy this equation are called...
i) Write the time-independent Scrödinger wave equation for helium (He) and H + 2 atoms. ii)...
i) Write the time-independent Scrödinger wave equation for helium (He) and H + 2 atoms. ii) If interactions between electrons are ignored in these equations, find the energy of the system in terms of electronvolt (eV). iii) Write the spin wave function of a system consisting of two electrons. iv) Specify how the angular momentum of an electron is defined according to classical mechanics, bohr atomic theory, and quantum mechanics, and the values it will take. note: Please show all...
-ax²12 directly into the Schroedinger equation, as broken down in the following steps. Show that the...
-ax²12 directly into the Schroedinger equation, as broken down in the following steps. Show that the energy of a simple harmonic oscillator in the n = 0 state is 1ho/2 by substituting the wave function wo = Ae First, calculate dvo/dx, using A, x, and a. dyo/dx = Second, calculate dvo/dx?, using A, x, and a. dyo/dx2 = Third, calculate a?x?wo-dayo/dx?, using A, x, and a. a3x240 - dạyo/dx? Fourth, calculate (a?x240-d2vo/dx2)/yo, using A, X, and a. (22x200-2vo/dx?)/- 1 Finally,...
Very confused by this problem, please help! Thanks! Ae-ax72 directly into the Schroedinger equation, as Show...
Very confused by this problem,
please help! Thanks!
Ae-ax72 directly into the Schroedinger equation, as Show that the energy of a simple harmonic oscillator in the n 0 state is 1ho/2 by substituting the wave function o broken down in the following steps. First, calculate dupo/dx, using A, x, and ?. duo/dx Second, calculate d2?0/dx, using A, x, and a. Third, calculate ?2x2PD-d2Wo/dx2, using A, x, and ?. Fourth, calculate (a2x2Wo-d2Wo/dx2)/40, using A, x, and ?. Finally, calculate E-[(c2x2Wo-d240/dx2)/Wo]h2/(2m), using...
Question 8 please 5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
[1] Determine all potentials V(r0,) for which it is possible to find solutions of the time-independent Schroedinger equation which are also eigenfunctions of the operator L. (Help: The operator e...
[1] Determine all potentials V(r0,) for which it is possible to find solutions of the time-independent Schroedinger equation which are also eigenfunctions of the operator L. (Help: The operator expression of the Hamiltonian for a particle of mass m in threedimensions is given (r))- 2m r ar2 1,2 2mr2
[1] Determine all potentials V(r0,) for which it is possible to find solutions of the time-independent Schroedinger equation which are also eigenfunctions of the operator L. (Help: The operator expression of...
17. Does the solution to the time-dependent Schrodinger equation for a free particle satisfy the classical...
17. Does the solution to the time-dependent Schrodinger equation for a free particle satisfy the classical nondispersive wave equation? Use mathematics to justify your answer.
Exercise: In this exercise, you will calculate various expectation values for different wave functions corresponding to...
Exercise: In this exercise, you will calculate various expectation values for different wave functions corresponding to different potentials. These distributions appear in Table VIII. Table VIII Infinite Square Well 1-D Hydrogen Atom Finite Smooth Well (ground state) (ground state) A Wave Function Un(x) = A sin 4.(x) = A x exp Wo(x) = cosh(kx) SMHB hint 17.17.1 to 17.19.1 18.76, 25.3 U(x) = _ kee? Potential U(X) = 0 ħ2 12 n2 17.27.1 to 17.33.7, 25.3 U. U(x) = -...
QM 30 Relating classical and quantum mechanics IV. Supplement: Highly-excited energy eigenstates A particle is in...
QM 30 Relating classical and quantum mechanics IV. Supplement: Highly-excited energy eigenstates A particle is in the potential well shown at right A. First, treat this problem from a purely elassical standpoint (assume the particle has enough energy to reach both regions) Give an example of a real physical situation that corresponds to this potential well. 1. region I | region II 2. In which region of the well would the particle have greater kinetic energy? Explain. 3. In which...
a) The wave-functions of the states [) and (o) are given by y(x) and (x), respectively....
a) The wave-functions of the states [) and (o) are given by y(x) and (x), respectively. Derive the expression for the inner product (4) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? b) Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states [4) and (o) are orthogonal: (14) = 0. (x) M Figure 1 c) Assume a particle has a wave-function y(x) sketched in Fig. 2....
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...
-ax²12 directly into the Schroedinger equation, as broken down in the following steps. Show that the energy of a simple harmonic oscillator in the n = 0 state is 1ho/2 by substituting the wave function wo = Ae First, calculate dvo/dx, using A, x, and a. dyo/dx = Second, calculate dvo/dx?, using A, x, and a. dyo/dx2 = Third, calculate a?x?wo-dayo/dx?, using A, x, and a. a3x240 - dạyo/dx? Fourth, calculate (a?x240-d2vo/dx2)/yo, using A, X, and a. (22x200-2vo/dx?)/- 1 Finally,...
Very confused by this problem,
please help! Thanks!
Ae-ax72 directly into the Schroedinger equation, as Show that the energy of a simple harmonic oscillator in the n 0 state is 1ho/2 by substituting the wave function o broken down in the following steps. First, calculate dupo/dx, using A, x, and ?. duo/dx Second, calculate d2?0/dx, using A, x, and a. Third, calculate ?2x2PD-d2Wo/dx2, using A, x, and ?. Fourth, calculate (a2x2Wo-d2Wo/dx2)/40, using A, x, and ?. Finally, calculate E-[(c2x2Wo-d240/dx2)/Wo]h2/(2m), using...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
[1] Determine all potentials V(r0,) for which it is possible to find solutions of the time-independent Schroedinger equation which are also eigenfunctions of the operator L. (Help: The operator expression of the Hamiltonian for a particle of mass m in threedimensions is given (r))- 2m r ar2 1,2 2mr2
[1] Determine all potentials V(r0,) for which it is possible to find solutions of the time-independent Schroedinger equation which are also eigenfunctions of the operator L. (Help: The operator expression of...
17. Does the solution to the time-dependent Schrodinger equation for a free particle satisfy the classical nondispersive wave equation? Use mathematics to justify your answer.
Exercise: In this exercise, you will calculate various expectation values for different wave functions corresponding to different potentials. These distributions appear in Table VIII. Table VIII Infinite Square Well 1-D Hydrogen Atom Finite Smooth Well (ground state) (ground state) A Wave Function Un(x) = A sin 4.(x) = A x exp Wo(x) = cosh(kx) SMHB hint 17.17.1 to 17.19.1 18.76, 25.3 U(x) = _ kee? Potential U(X) = 0 ħ2 12 n2 17.27.1 to 17.33.7, 25.3 U. U(x) = -...
QM 30 Relating classical and quantum mechanics IV. Supplement: Highly-excited energy eigenstates A particle is in the potential well shown at right A. First, treat this problem from a purely elassical standpoint (assume the particle has enough energy to reach both regions) Give an example of a real physical situation that corresponds to this potential well. 1. region I | region II 2. In which region of the well would the particle have greater kinetic energy? Explain. 3. In which...
a) The wave-functions of the states [) and (o) are given by y(x) and (x), respectively. Derive the expression for the inner product (4) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? b) Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states [4) and (o) are orthogonal: (14) = 0. (x) M Figure 1 c) Assume a particle has a wave-function y(x) sketched in Fig. 2....
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...
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