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• ### Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a)...

Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a) State the time-independent Schrödinger equation within the region 0<x<L for a particle with positive energy E 2 marks] (b) The wavefunction for 0<x< L can be written in the general form y(x) = Asin kx + B cos kx. Show that the normalised wavefunction for the 1D infinite potential well becomes 2sn'n? ?snT/where ( "1,2,3 ! where ( n = 1,2,5, ). [4 marks]...

• ### 2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, w...

2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...

• ### Questions 1 - 5 deal with a particle in a one-dimensional infinite square well of width...

Questions 1 - 5 deal with a particle in a one-dimensional infinite square well of width a where 0, 0 SX Sa V(x) = 100, Otherwise. The stationary states are Pn(x) = sin(**) with energies En = "forn = 1,2,3.. Question 1 (14 pts) Which of the following is correct? A. The Hilbert space for this system is one dimensional. B. The energy eigenstates of the system form a ID Hilbert space. C. Both A and B are correct. D....

• ### 4) A particle in an infinite square well 0 for 0

4) A particle in an infinite square well 0 for 0

• ### A particle is trapped in a one-dimensional potential energy well given by: 100 x < 0...

A particle is trapped in a one-dimensional potential energy well given by: 100 x < 0 0 < x <L U(x) = L < x < 2L (20. x > 2L Consider the case when U, < E < 20., where E is the particle energy. a. Write down the solutions to the time-independent Schrödinger equation for the wavefunction in the four regions using appropriate coefficients. Define any parameters used in terms of the particles mass m, E, U., and...

• ### 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?

• ### 3. For a particle moving in an infinite, one-dimensional, symmetric square well of width 2a, show...

3. For a particle moving in an infinite, one-dimensional, symmetric square well of width 2a, show that the (normalized) wave functions are of the form ?-kx).va. cos?x): "-1. 3.5 ,.. COS ? -?? r")(x)=?sin n-r | ; n-2, 4, 6 Express the state ?(x)=N sin,(rx/a) as a linear superposition eigenstates, and find its normalization constant N. of the above HINT sin39-3sin ?-4sin'?

• ### At time t = 0, a mass-m particle in a one-dimensional potential well is in a...

At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6] At time t = 0, a...

• ### 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...

• ### Parity (please answer from part a to part d) Consider Infinite Square Well Potential, V(x) =...

Parity (please answer from part a to part d) Consider Infinite Square Well Potential, V(x) = 0 for |x| < 1/2a and V(x) = infinity for |x| > 1/2a a) Find energy eigenstates and eigenvalues by solving eigenvalue equation using appropriate boundary conditions. And show orthogonality of eigenstates. For rest of part b to part d please look at the image below: Problem 1 . Parity Consider an infinite square well potential, V(x) = 0 for lxl 〈 a and...