A. Momentum space We define the momentum space wave function φ(p) as where Ψ(x)is a solution...
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0, otherwise where a and A are positive real numbers. The particle is in a zero (or constant) potential environment since it is a free particle a) Determine A from normalization. b) Determine φ(p) = Φ(p,0), the time-zero momentum representation of the particle state. What is Φ(p,t)? Sketch φ(p). Locate the global maximum and the zeros of φ(p). Give the expression for the zeros (i.e.,...
Show that momentum space is equivalent to position space knowing that the operator X̂=i(hbar)(∂/∂p). ( ∫(-∞ --> ∞) Ψ•(x) x Ψ(x)dx = ( ∫(-∞ --> ∞) Φ•(p) (i(hbar)(∂/∂p)) Φ(p)dp Please show detailed steps. Thank you.
The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p Ψ & ΕΨ ) as to verify the following pshk and Eshω Schrodinger sequation...-Nay equation... Ew andthen wufythefollowing: b) Substitute w into 2m ax E-Pi 2m The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p...
Consider a particle of mass m that is described by the wave function (x, t) = Ce~iwte-(x/l)2 where C and l are real and positive constants, with / being the characteristic length-scale in the problem Calculate the expectation values of position values of 2 and p2. and momentum p, as well as the expectation Find the standard deviations O and op. Are they consistent with the uncertainty principle? to be independent What should be the form of the potential energy...
3. Consider the wave function ψ(x)- 슬 읔 ets, where σ s a real valued constant (a) Calculate the expectation value of K). K (b) Estimate the uncertainty Δ.r and Ap using Δ.1-V (.12)-(A)2. 4. Consider the eigenfunctions of the moment uni operator y p r her (a) Show that p,(r) is an eigenfunction of p with an cigenvalue p. (b) Find the coeflicients. w, in the espansion of (r)( upypp ) using the momentum eigenfunctions.
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish by calculating the following quantities. (Use the following as necessary: A, K, x, ,t, h, and m.) momentum Need Help?Read ItTalk to a Tutor 9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish...
Let H be a complex Hilbert space. 6. (a) Let φ, ψ E H \ {0} . Define the linear operator T on H by Using the Cauchy-Schwarz inequality, show that llll = Hell ll [4 marks] (b) A bounded linear operator A is said to have rank one if there exists v e H [0 such that for any u E H we have Au cu, where cu E C is a constant depending on u. (i) Show that...
*Please, answer all the literals and be detailed with the answer (do all the procedure and calculations) *Do it with a clear letter Homework (scattering) 1. Consider the time dependent Schrödinger equation written in the form where 0 2mo As it is well known the temporal evolution of a wave function ψ( t) known at a specific time t is uniquely determined for all future times t, > t as well as for all past times t' < t. Moreover,...
1. Problems on unitary operators. For a function f(r) that can be expanded in a Taylor series, show that Here a is a constant, and pis the momentum operator. The exponential of an operator is defined as ea_ ??? i,O" Verify that the unitary operator elo/h can be constructed as follows (Hint: Notice that f(x +a) (al) and eohf())) e Prove that Here is the position operator. (Hint: You may work in the momentum space, in which p = p...
additional info 17.1) Show that the retarded field propagator for a free particle in momentum space and the time domain: given by θ(te-ty)e-i(Epte-Eqty's(3) (p-q) 17.1 The field propagntor in outline e field propagator in outline 155 he field propagator involves a simple thought experi- our interacting system in its ground state, which we interactin ent. We start with denote w 12). The thought experiment works as follows: we introdu extra particle of our choice the system. point ( in anni...