Homework Answers
47 (a) A particle is confined inside a rectangular box with sides of length a, a,...
47 (a) A particle is confined inside a rectangular box with sides of length a, a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy b) Now assume the rectangular box has sides of length a, 2a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy.
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...
II.6. The wave function of a particle in a 1D rigid box (infinite potential well) of...
II.6. The wave function of a particle in a 1D rigid box (infinite potential well) of length L is: v, 8, 1) = sin(x)e-En/5). n = 1,2,3... What is the probability density of finding the particle in its 2nd excited state?
For the particle with a mass m in a box with a length of 1 nm,...
For the particle with a mass m in a box with a length of 1 nm, a) Write Schrodinger’s equation b) Write the integral expression for the probability of finding the first excited state between 0.1 nm and 0.5 nm
A particle of mass m moves in one dimension. Its potential energy is given by U(x)...
A particle of mass m moves in one dimension. Its potential energy is given by U(x) = -Voe-22/22 where U, and a are constants. (a) Draw an energy diagram showing the potential energy U(). Choose some value for the total mechanical energy E such that -U, < E < 0. Mark the kinetic energy, the potential energy and the total energy for the particle at some point of your choosing. (b) Find the force on the particle as a function...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, a...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
1- An infinite potential box containing a particle of mass m is Perturbed as shown below....
1- An infinite potential box containing a particle of mass m is Perturbed as shown below. i. Find the 1st order correction to the G.S. Energy of the particle. (10 points) 0.5 -0.5. E°(n=1) ii- Find the 1st order correction (the 1st non-zero term) to the G.S. Wave Function of the particle. (15 points). iii- Find the 2nd order correction to the G.S. Energy of the particle. (15 points)
A particle of mass m is attached to a fixed point in space by a massless rigid rode of length a a...
A particle of mass m is attached to a fixed point in space by a massless rigid rode of length a and can freely rotate about this point. Find the quan- tum energy levels of the system. What is the degeneracy of each energy level (i.e. how many different quantum states have given energy)? Compare to the one of hydrogen atom
A particle of mass m and energy E moving along the x axis is subjected to a potential energy function U(x).
\((25\) marks) A particle of mass \(m\) and energy \(E\) moving along the \(x\) axis is subjected to a potential energy function \(U(x) .\) (a) Suppose \(\psi_{1}(x)\) and \(\psi_{2}(\mathrm{x})\) are two wave functions of the system with the same energy \(E .\) Derive an expression to relate \(\psi_{1}(x), \psi_{2}(x)\), and their derivatives. (b) By requiring the wave functions to vanish at infinity, show that \(\psi_{1}(x)\) and \(\psi_{2}(x)\) can at most differ by a multiplicative constant. Hence, what conclusion can you...
2. (a) When a particle of mass 1.0 x 10-26 g in a one-dimensional box goes...
2. (a) When a particle of mass 1.0 x 10-26 g in a one-dimensional box goes from the n=3 level to n=1 level, it emits a radiation with frequency 5.0 x 1014 Hz. Calculate the length of the box. (b) Suppose that an electron freely moves around inside of a three-dimensional rectangular box with dimensions of 0.4 nm (width), 0.4 nm (length), and 0.5 nm (height). Calculate the frequency of the radiation that the electron would absorb during its transition...
47 (a) A particle is confined inside a rectangular box with sides of length a, a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy b) Now assume the rectangular box has sides of length a, 2a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy.
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...
II.6. The wave function of a particle in a 1D rigid box (infinite potential well) of length L is: v, 8, 1) = sin(x)e-En/5). n = 1,2,3... What is the probability density of finding the particle in its 2nd excited state?
A particle of mass m moves in one dimension. Its potential energy is given by U(x) = -Voe-22/22 where U, and a are constants. (a) Draw an energy diagram showing the potential energy U(). Choose some value for the total mechanical energy E such that -U, < E < 0. Mark the kinetic energy, the potential energy and the total energy for the particle at some point of your choosing. (b) Find the force on the particle as a function...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
1- An infinite potential box containing a particle of mass m is Perturbed as shown below. i. Find the 1st order correction to the G.S. Energy of the particle. (10 points) 0.5 -0.5. E°(n=1) ii- Find the 1st order correction (the 1st non-zero term) to the G.S. Wave Function of the particle. (15 points). iii- Find the 2nd order correction to the G.S. Energy of the particle. (15 points)
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