In class, we considered a box with walls at x = 0 and x = L. Now consider a box with width L but centered at x = 0, so that it extends from x = L/2 to x = L/2 as shown in the figure.
In class, we considered a box with walls at \(x=0\) and \(x=L\). Now consider a box with width \(L\) but centered at \(x=0\), so that it extends from \(x=-L / 2\) to \(x=L / 2\) as shown in the figure. Note that this box is symmetric about \(x=0 .\) (a) Consider possible wave functions of the form \(\psi(x)=A \sin k x\). Apply the boundary conditions at the wall to obtain the allowed energy levels.
(b) Another set of possible wave functions are functions of the form \(\psi(x)=A \cos k x\). Apply the boundary conditions at the wall to obtain the allowed energy levels. (c) Compare the energies obtained in parts (a) and (b) to the set of energies given in the lecture note (Ch.8 pg 7). (d) An odd function \(f\) satisfies the condition \(f(x)=-f(-x),\) and an even function \(g\) satisfies \(g(x)=g(-x)\). Of the wave functions from parts (a) and (b), which are even and which are odd?
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In class, we considered a box with walls at x = 0 and x = L. Now consider a box with width L but centered at x = 0, so that it extends from x = L/2 to x = L/2 as shown in the figure.
Consider a potential well defined as U(x) = for x < 0, U(x) = 0 for 0 < x < L, and U(x) = U0 > 0 for x > L (see the following figure).
Consider a potential well defined as \(U(x)=\infty\) for \(x<0, U(x)=0\)for \(0<x<L,\)and \(U(x)=U_{0}>0\) for \(x>L\) (see the following figure). Consider a particle with mass \(m\) and kinetic energy \(E<U_{0}\)that is trapped in the well. (a) The boundary condition at the infinite wall ( \(x=\)0) is \(\psi(x)=0\). What must the form of the function \(\psi(x)\) for \(0<x<L\)be in order to satisfy both the Schrödinger equation and this boundary condition? (b) The wave function must remain finite as \(x \rightarrow \infty\). What must...
1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential....
1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential. Sketch what the wave functions would look like for α = 0 for the ground state and the 1st excited state. Write down a formula for all of the bound state energies for α = 0 (no derivation necessary). a) b) Break up the x axis into regions where the Schrödinger equation is easy to solve. Guess solutions in these regions and plug them...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
In class we looked at the example of the potential energy step seen below (where E...
In class we looked at the example of the potential energy step seen below (where E > U_0). We wrote down the wave functions in complex exponential form as seen below: psi _0 (x) = A' e^i K_0 x + B' e^-i K_0 x x < 0 psi _1 (x) = C' e^i K_1 x + D' e^-i K_1 x x > 0 a) Assume the particles are incident on the barrier from the left, which coefficient can be set...
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) =...
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
Question 8 please 5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
In class we considered quantum tunneling of a particle of energy Eo through a barrier of...
In class we considered quantum tunneling of a particle of energy Eo through a barrier of potential Vofor Vo > Eo. Here we focus on two aspects of the problem we ignored in class. In order to simplify we will only consider the initial first half of the barrier as shown below RegionI xS0 Regionx 20 Il There are two cases to consider: Eo< Vo Considered in class E>Vo Not considered in class Here we will focus on the second...
3 Casimir effect c. All vacuum energies considered so far are infinite. In reality, the metal wal...
3 Casimir effect c. All vacuum energies considered so far are infinite. In reality, the metal walls are conduc tors only at finite frequencies, and thus do not impose boundary conditions at infinite frqu cies. (The cutoff is given roughly by the plasma frequencywp Thus, we need (and should) not consider infinite frequencies in our equations. To remove We will derive the Casimir effect in thee d sions, making use of the Euler-Maclaurin formula imen- them, we introduce a cutoff...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) =...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
A plucked string. Figure 2 shows the initial position function f (x) for a stretched string (of lengt L) that is set in motion by moving it at midpoint x = ! aside the distance 1 bL and releasing...
A plucked string. Figure 2 shows the initial position function f (x) for a stretched string (of lengt L) that is set in motion by moving it at midpoint x = ! aside the distance 1 bL and releasing it from rest time t = 0 f (x) Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the problem. [ Hints: f(x) = bx...
1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential. Sketch what the wave functions would look like for α = 0 for the ground state and the 1st excited state. Write down a formula for all of the bound state energies for α = 0 (no derivation necessary). a) b) Break up the x axis into regions where the Schrödinger equation is easy to solve. Guess solutions in these regions and plug them...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
In class we looked at the example of the potential energy step seen below (where E > U_0). We wrote down the wave functions in complex exponential form as seen below: psi _0 (x) = A' e^i K_0 x + B' e^-i K_0 x x < 0 psi _1 (x) = C' e^i K_1 x + D' e^-i K_1 x x > 0 a) Assume the particles are incident on the barrier from the left, which coefficient can be set...
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
In class we considered quantum tunneling of a particle of energy Eo through a barrier of potential Vofor Vo > Eo. Here we focus on two aspects of the problem we ignored in class. In order to simplify we will only consider the initial first half of the barrier as shown below RegionI xS0 Regionx 20 Il There are two cases to consider: Eo< Vo Considered in class E>Vo Not considered in class Here we will focus on the second...
3 Casimir effect c. All vacuum energies considered so far are infinite. In reality, the metal walls are conduc tors only at finite frequencies, and thus do not impose boundary conditions at infinite frqu cies. (The cutoff is given roughly by the plasma frequencywp Thus, we need (and should) not consider infinite frequencies in our equations. To remove We will derive the Casimir effect in thee d sions, making use of the Euler-Maclaurin formula imen- them, we introduce a cutoff...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
A plucked string. Figure 2 shows the initial position function f (x) for a stretched string (of lengt L) that is set in motion by moving it at midpoint x = ! aside the distance 1 bL and releasing it from rest time t = 0 f (x) Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the problem. [ Hints: f(x) = bx...
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