show that psi(z,t) = Asin^2*4*pi(t+z) is a solution of one dimensional wave equation
show that psi(z,t) = Asin^2*4*pi(t+z) is a solution of one dimensional wave equation
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show that psi(z,t) = Asin^2*4*pi(t+z) is a solution of one
dimensional wave equation
4. a) The one dimensional wave equation for the variable y(z, t) can be written as:...
4. a) The one dimensional wave equation for the variable y(z, t) can be written as: azz czacz where w is the angular frequency (rad/s), k is the wavenumber (rad/m) t is time (s). Show that y(z, t) = 12sin(wt + kz) - 24sin(wt - kz) is a valid solution. (15 marks) b) If a string is fixed at z = Om and at z = 2.4m, and its displacement when vibrating in its fundamental mode is given by: y(z,...
The one-dimensional Schrindinger wave equation for a particle in a potential field
The one-dimensional Schrindinger wave equation for a particle in a potential field \(V=\) \(\frac{1}{2} k x^{2}\) is$$ -\frac{h^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+\frac{1}{2} k x^{2} \psi=E \psi(x) $$(a) Lsing \(\xi=\alpha x\) and a constant \(\lambda\), we have$$ a=\left(\frac{m k}{A^{2}}\right)^{1 / 4}, \quad A=\frac{2 L}{A}\left(\frac{m}{k}\right)^{1 / 2} $$show that$$ \frac{d^{2} y(\xi)}{d \xi^{2}}+\left(\lambda-\xi^{2}\right) \psi(\xi)=0 $$(b) Substituting$$ \psi(\xi)=y(\xi) e^{2} / 2 $$show that \(y(t)\) satisfies the Hermite di fferential equation.
Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou...
Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou ot (4) possible , some work has been hard to follow Thanks! a(z,0) = e-r2 (a) Determine the solution u(r,t) of (4) (b) Sketch the solution in the xu-plane at t = 0, t = 1 , and (c) Which direction does the wave travel? 2
3. An atom is in a time-independent one-dimensional potential well. The system's spatial wave function is ψ(x)-Asin(2mz/L) for 0 < x < L and zero for all other z. What is the average of...
3. An atom is in a time-independent one-dimensional potential well. The system's spatial wave function is ψ(x)-Asin(2mz/L) for 0 < x < L and zero for all other z. What is the average of the position operator? For which o is the probability that the atom is located in the interval [xo 0.01L, o 0.01L] largest?
3. An atom is in a time-independent one-dimensional potential well. The system's spatial wave function is ψ(x)-Asin(2mz/L) for 0
check whether the function E(x,t)= Asin(kx^2-wt^2) satisfies the wave equation. if so, find the wave speed....
check whether the function E(x,t)= Asin(kx^2-wt^2) satisfies the wave equation. if so, find the wave speed. if not explain
2. Show that the function w = ln(2x + 2ct) is a solution of the one-dimensional...
2. Show that the function w = ln(2x + 2ct) is a solution of the one-dimensional wave equation, aw aw at2 Әr2
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation,...
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation, then c ψ(x), where c is an arbitrary complex constant, is also a solution.
(10 points) Show that for a one-dimensional square integrable wave packet given by the wave function...
(10 points) Show that for a one-dimensional square integrable wave packet given by the wave function Ψ(x,t), the following relation is true: j(x) da p) 7n where j z is the probability current defined in the previous question. Use the fact that ψ(z, t goes to zero as x → ±00 and also use integration by parts.
4.(10pts) Write Laplaces' equation in cylindricaol co-ordinates(p527 ex.3,use pinstead ofr) Assume the solution, e, φ, z), n can be written φ (p, φ, z)s u(p, φ)e-kz and Show that the equation...
4.(10pts) Write Laplaces' equation in cylindricaol co-ordinates(p527 ex.3,use pinstead ofr) Assume the solution, e, φ, z), n can be written φ (p, φ, z)s u(p, φ)e-kz and Show that the equation for u is the two dimensional wave equation; Written in polar co-ordinates:xpcosp,y psinp For a plane wave traveling in a direction defined by:4-kcosce, ky-kinα Show that the plane wave solution can be written; look for a solution u z(x)en (2-n212,-0 And the equation for Z, is Bessels equation:Zh "x2...
PDE question Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt...
PDE
question
Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
4. a) The one dimensional wave equation for the variable y(z, t) can be written as: azz czacz where w is the angular frequency (rad/s), k is the wavenumber (rad/m) t is time (s). Show that y(z, t) = 12sin(wt + kz) - 24sin(wt - kz) is a valid solution. (15 marks) b) If a string is fixed at z = Om and at z = 2.4m, and its displacement when vibrating in its fundamental mode is given by: y(z,...
Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou ot (4) possible , some work has been hard to follow Thanks! a(z,0) = e-r2 (a) Determine the solution u(r,t) of (4) (b) Sketch the solution in the xu-plane at t = 0, t = 1 , and (c) Which direction does the wave travel? 2
3. An atom is in a time-independent one-dimensional potential well. The system's spatial wave function is ψ(x)-Asin(2mz/L) for 0 < x < L and zero for all other z. What is the average of the position operator? For which o is the probability that the atom is located in the interval [xo 0.01L, o 0.01L] largest?
3. An atom is in a time-independent one-dimensional potential well. The system's spatial wave function is ψ(x)-Asin(2mz/L) for 0
2. Show that the function w = ln(2x + 2ct) is a solution of the one-dimensional wave equation, aw aw at2 Әr2
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation, then c ψ(x), where c is an arbitrary complex constant, is also a solution.
(10 points) Show that for a one-dimensional square integrable wave packet given by the wave function Ψ(x,t), the following relation is true: j(x) da p) 7n where j z is the probability current defined in the previous question. Use the fact that ψ(z, t goes to zero as x → ±00 and also use integration by parts.
4.(10pts) Write Laplaces' equation in cylindricaol co-ordinates(p527 ex.3,use pinstead ofr) Assume the solution, e, φ, z), n can be written φ (p, φ, z)s u(p, φ)e-kz and Show that the equation for u is the two dimensional wave equation; Written in polar co-ordinates:xpcosp,y psinp For a plane wave traveling in a direction defined by:4-kcosce, ky-kinα Show that the plane wave solution can be written; look for a solution u z(x)en (2-n212,-0 And the equation for Z, is Bessels equation:Zh "x2...
PDE
question
Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
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