Question 19 The wave function of a quantum particle of mass m is where A. B, and кare constants. (a) (x) = A cos(kx) + B sin(kx) Assuming the particle is free (U= 0), show that (x) is a solution of the Schrödinger equation. (b) Find the corresponding energy E of the particle.
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- Check your Understanding (a) Consider an infinite square well with wall boundaries x=0 and x=L. What is the of finding a quantum panicle in its state somewhere between x=0 and x=L/4? (b) Repeat question (a) for a classical panicle.Check Your Understanding A sodium atom nukes a transition from the first excited state the wound state, emitting a 589.0-nm photon with energy 2.105 eV. If the lifetime of this excited state is 16108s, what is the uncertainty in energy of this state? What is width of the corresponding line?A particle of mass m is moving in an infinite 1D quantum well of width L. y,(x) = J? sinx. sin nAx L (a) How much energy must be given to the particle so it can transition from the ground state to the second excited state? (b) If the particle is in the first excited state, what is the probability of finding the particle between x = and x = ;? 2.
- What is the probability of measuring the energy En of a particle in the combination of the states below? p(x, t) = Eann (x, t)In quantum mechanics. Prove thatQuestion 1 a) Write down the one-dimensional time-dependent Schro ̈dinger equation, for a particle de- scribed by a wavefunction Ψ(t, x) in a potential V (t, x). b) State the condition that the potential must obey in order to derive the time-independent Schro ̈dinger equation from the time-dependent equation. c) Write down a mathematical expression for the wavefunction of a particle in an energy eigen- state of such a potential. Your answer should be given in terms of the spatially dependent part of the wavefunction, ψ(x), and a factor that depends on the energy eigenvalue of the particle, E. d) Using your answers to parts a)-c) of this question, derive the one-dimensional time-independent Schro ̈dinger equation for the particle described by the wavefunction ψ(x).
- .An electron is confined in a cubical box ofside 1Å. Find the momentum and energy ofthe electron in the ground state and the first excited state.A quantum mechanical particle of mass m moves in a 1D potential where a) Estimate the ground state energy of the particle. b) Sketch the wave function to the best of your ability.It can be shown that the allowed energies of a particle of mass m in a two-dimensional square box of sided L are Enl =h2/8mL2 (n2 + l2)The energy depends on two quantum numbers, n and l, both of which must have an integer value 1, 2, 3,........a. What is the minimum energy for a particle in a twodimensional square box of side L?b. What are the five lowest allowed energies? Give your values as multiples of Emin .
- Q3. Consider an infinite potential well of width d. In transitions between neighboring values of n, particles of mass that is in a position state as: |1 TX -iwot sin 2πχ sin d f(x. t) = + -iwit d (a) Proof that f(x.t) is still normalized for all value of t. (b) Find the probability distribution P(x.t) = |f(x.t)|²Advanced Physics A single-particle system exists in free space, with d being a real constant, in the state described by the momentum wave function @(p)= N exp(=d*(p-ħk) ) a) What is the meaning of the one to-one dimensioning of and find the constant? b) Show that it achieves the minimum pesition-momentum uncertainty for this wave function.An electron trapped in a one-dimensional infinitely deep potential well with a width of 250 pm is excited from the first excited state to the third excited state. What energy must the electron acquire for this quantum jump to occur? The electron then emits a photon and transitions to the ground state. Determine the wavelength and momentum of the emitted photon.
