Quantum Mechanics Wave Function
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Questions and Answers

What is the name of the quantum mechanical effect that allows particles to penetrate classically impenetrable barriers?

  • Classical penetration
  • Tunneling effect (correct)
  • Wave-particle duality
  • Barrier reflection

    • What is the region that particles cannot enter according to classical physics?

  • Region x < 0
  • Region x > a
  • Region x < a (correct)
  • Region x = a

    • What is the application of the tunneling effect in electronic devices?

  • Charge transport (correct)
  • Energy storage
  • Signal amplification
  • Voltage regulation

    • What is the mathematical symbol for the potential barrier height?

    <p>V0</p> Signup and view all the answers

    What is the name of the equation that describes the transmission coefficient T?

    <p>Equation (4.64)</p> Signup and view all the answers

    What is the physical quantity represented by the symbol 'm' in the equations?

    <p>Mass of the particle</p> Signup and view all the answers

    What is the process described by the equation (4.67)?

    <p>Tunneling through a square barrier</p> Signup and view all the answers

    What is the physical significance of the parameter 'D' in the equations?

    <p>A dimensionless parameter</p> Signup and view all the answers

    What is the formula to calculate the transmission probability corresponding to V(xi)?

    <p>Ti = exp(-2<em>m</em>V(xi) / h**2</p> Signup and view all the answers

    What is the transmission probability for the general potential of Figure 4.5?

    <p>T = lim(N→∞) ∏[exp(-2<em>m</em>V(xi) / h**2]</p> Signup and view all the answers

    What is the condition for the approximation leading to equation (4.71) to be valid?

    <p>The potential V(x) is a smooth, slowly varying function of x.</p> Signup and view all the answers

    What is the limiting process involved in equation (4.71)?

    <p>N → ∞</p> Signup and view all the answers

    What is the variable of integration in equation (4.71)?

    <p>x</p> Signup and view all the answers

    What are the two equations that lead to the expressions for C and D?

    <p>Equations (4.50) and (4.51)</p> Signup and view all the answers

    What is the relation between the transmission probability and the potential V(x)?

    <p>The transmission probability is exponentially related to the potential V(x).</p> Signup and view all the answers

    What is the result of dividing equations (4.50) and (4.51) by A?

    <p>Equations (4.55) and (4.56)</p> Signup and view all the answers

    In what chapter is the validity of the approximation leading to equation (4.71) discussed?

    <p>Chapter 9</p> Signup and view all the answers

    What is the expression for B/A in terms of k1, k2, and a?

    <p>(k12 - k22) / (k1 k2)</p> Signup and view all the answers

    What is the physical significance of the transmission probability?

    <p>It represents the probability of transmission.</p> Signup and view all the answers

    What is the expression for E/A in terms of k1, k2, and a?

    <p>(k22 - k12) / (k1 k2)</p> Signup and view all the answers

    What is the expression for the reflection coefficient R?

    <p>sinh2(k2 a) / (k12 + k22)</p> Signup and view all the answers

    What is the expression for the transmission coefficient T?

    <p>4 cosh2(k2 a) / (k1 k2)</p> Signup and view all the answers

    What is the relationship between R and T?

    <p>R = T / 4</p> Signup and view all the answers

    What is the rewritten expression for T in terms of k1, k2, and a?

    <p>1 / (1 + sinh2(k2 a))</p> Signup and view all the answers

    Is the transmission coefficient T finite?

    <p>Yes</p> Signup and view all the answers

    What is the relationship between cosh2(k2 a) and sinh2(k2 a)?

    <p>cosh2(k2 a) = 1 + sinh2(k2 a)</p> Signup and view all the answers

    What is the phenomenon where particles can pass through a potential barrier?

    <p>Tunneling</p> Signup and view all the answers

    What is the significance of the points x1 and x2 in Figure 4.5?

    <p>They are the classical turning points</p> Signup and view all the answers

    What is the purpose of the Wentzel-Kramers-Brillouin (WKB) method?

    <p>To provide an approximation for the tunneling probability</p> Signup and view all the answers

    What is the assumption made in the crude approximation to obtain the transmission coefficient?

    <p>The potential V xi is approximately constant over small intervals xi</p> Signup and view all the answers

    What is the expression for the transmission coefficient given in the text?

    <p>T_r = exp(-2/h * dx (2m(V(x) - E))) from x1 to x2</p> Signup and view all the answers

    What is the significance of the region x1 x x2?

    <p>It is a classically forbidden region</p> Signup and view all the answers

    What is the purpose of dividing the classically forbidden region into small intervals xi?

    <p>To approximate the potential V xi by a square potential barrier</p> Signup and view all the answers

    What is the condition for the intervals xi to be valid?

    <p>The intervals xi should be small enough</p> Signup and view all the answers

    What is the shape of the potential well in the given problem?

    <p>Asymmetric</p> Signup and view all the answers

    What is the momentum of the particle according to classical mechanics?

    <p>Constant and changing direction</p> Signup and view all the answers

    What is the expected energy spectrum of the particle according to quantum mechanics?

    <p>Discrete and nondegenerate</p> Signup and view all the answers

    Why is the wave function zero outside the boundary?

    <p>Because the potential is infinite outside the region</p> Signup and view all the answers

    What is the general form of the wave function inside the well?

    <p>A sin(kx) + B cos(kx)</p> Signup and view all the answers

    What is the condition that determines the energy of the particle?

    <p>kn = nπ</p> Signup and view all the answers

    What is the energy of the nth state of the particle?

    <p>E_n = h^2n^2/2ma^2</p> Signup and view all the answers

    Why is the energy spectrum discrete?

    <p>Because the particle is confined to a limited region of space</p> Signup and view all the answers

    Study Notes

    The Continuity Conditions of the Wave Function

    • The continuity conditions of the wave function and its derivative at x = 0 and x = a yield four equations (4.50-4.53) that relate the coefficients A, B, C, and D.
    • Solving these equations, we can express C and D in terms of A and E (4.54).

    The Coefficients R and T

    • The coefficients R and T can be obtained by solving the equations for B/A and E/A (4.57-4.58).
    • The coefficients R and T are related to the transmission and reflection probabilities of the particles.

    The Tunneling Effect

    • The tunneling effect is a quantum mechanical phenomenon where particles can tunnel through classically impenetrable barriers.
    • This effect is due to the wave aspect of microscopic objects and has important applications in various branches of modern physics.

    The Transmission Coefficient

    • The transmission coefficient T can be rewritten in terms of the energy E and the potential V0 (4.63).
    • The transmission coefficient T can be expressed in terms of the sinh function (4.65).
    • The transmission coefficient T can be approximated using the WKB method (4.69).

    The Potential Barrier

    • The potential barrier can be approximated by dividing the classically forbidden region into small intervals and using a square potential barrier for each interval (4.70).
    • The transmission probability for the general potential can be obtained by taking the limit of the product of the transmission probabilities for each interval (4.71).

    The Infinite Square Well Potential

    • The infinite square well potential is a potential where the particle is confined to move inside a well of infinite depth (4.72).
    • The wave function of the particle must be zero outside the boundary of the well.
    • The solutions to the Schrödinger equation are sine and cosine functions (4.74).
    • The energy is quantized, and only certain values are permitted (4.76).

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    Description

    Deriving equations for wave function and its derivative in quantum mechanics, applying continuity conditions at specific points.

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