Quantum Mechanics: Finite Square Well Potential
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Questions and Answers

What is the well width at which n states are allowed in a bound state?

  • RH = √(nh²/V₀) / (2ma²)
  • RH = √(2n²h²/V₀) / (2ma²) (correct)
  • RH = √(2n²h²/V₀) / (ma²)
  • RH = √(2nh²/V₀) / (2ma²)

    • What is the sequence of states in the spectrum of a bound state?

  • Even, even, odd, odd, ...
  • Odd, even, odd, even, ...
  • Even, odd, even, odd, ... (correct)
  • Odd, odd, even, even, ...

    • What is the parity of the first excited state?

  • Even
  • Odd (correct)
  • It depends on the value of n
  • Neither even nor odd

    • What is the parity of the ground state?

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

    How many bound states are there in the given system?

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

    What is the value of n for the second excited state?

    <p>2</p> Signup and view all the answers

    What type of eigenfunctions do symmetric one-dimensional Hamiltonians have?

    <p>Either even or odd</p> Signup and view all the answers

    What is the condition required to determine the eigenvalues?

    <p>The continuity conditions at x = a/2</p> Signup and view all the answers

    What can be said about the solutions of (4.90) to (4.92)?

    <p>They are either symmetric or antisymmetric</p> Signup and view all the answers

    What is the graphical representation of the solutions for the finite square well potential?

    <p>The intersections of R^2 :n2 with :n tan :n and :n cot :n</p> Signup and view all the answers

    What happens to the physically unacceptable solutions for large values of x?

    <p>They grow exponentially</p> Signup and view all the answers

    What is the condition for x in equation (4.93)?

    <p>x &lt; a/2</p> Signup and view all the answers

    What is the value of R when there is only one bound state?

    <p>R = 1</p> Signup and view all the answers

    What is the expression for Oa(x) in equation (4.95)?

    <p>Csin(k2x) - De^(k1x)</p> Signup and view all the answers

    What is the value of n corresponding to the bound state when R = 1?

    <p>n = 0</p> Signup and view all the answers

    What is the equation that determines the energy of the bound state?

    <p>:0 tan :0 = 1 + :02</p> Signup and view all the answers

    What is the condition for x in equation (4.94)?

    <p>x &gt; a/2</p> Signup and view all the answers

    What is the numerical solution of cos2 :0 = :02?

    <p>:0 = 0.73909</p> Signup and view all the answers

    What is the expression for Os(x) in equation (4.96)?

    <p>Bcos(k2x) - De^(k1x)</p> Signup and view all the answers

    What is the expression for the energy of the bound state?

    <p>ma^2 E0 = 0.73909 h^2 / (2m)</p> Signup and view all the answers

    What is the condition for having only one bound state?

    <p>R = 1</p> Signup and view all the answers

    What is the expression for R^2 in terms of m, V0, and h?

    <p>R^2 = m^2 V0 / (h^2)</p> Signup and view all the answers

    What is the form of the potential in the Hamiltonian described?

    <p>V(x) = 1/2 mω²x²</p> Signup and view all the answers

    What is the main challenge in solving the time-independent Schrödinger equation for the given Hamiltonian?

    <p>The equation is difficult to solve due to the complexity of the potential</p> Signup and view all the answers

    What is the purpose of the ladder or algebraic method in this context?

    <p>To find the energy eigenvalues and eigenstates of the Hamiltonian</p> Signup and view all the answers

    Why is the ladder or algebraic method preferred over the analytic method?

    <p>It is more straightforward, more elegant, and simpler</p> Signup and view all the answers

    What is the role of the creation and annihilation operators in the ladder or algebraic method?

    <p>They are used to express the various quantities in terms of matrices</p> Signup and view all the answers

    What is the advantage of the matrix formulation in the ladder or algebraic method?

    <p>It is more straightforward and simpler</p> Signup and view all the answers

    What is the condition for the energy spectrum to be continuous and doubly-degenerate?

    <p>E &gt; V0</p> Signup and view all the answers

    What happens to the particle's momentum between -a/2 and a/2?

    <p>It increases</p> Signup and view all the answers

    What is the value of R (reflection coefficient) in the region x < -a/2?

    <p>0</p> Signup and view all the answers

    What is the direction of the incident wave?

    <p>From left to right</p> Signup and view all the answers

    What is the range of x where the particle slows down?

    <p>x &gt; a/2</p> Signup and view all the answers

    What is the condition for the energy spectrum to be discrete and non-degenerate?

    <p>E &lt; V0</p> Signup and view all the answers

    What is the momentum of the particle initially?

    <p>$ ext{\sqrt}{2mE}$</p> Signup and view all the answers

    What is the value of T (transmission coefficient) in the region x < -a/2?

    <p>1</p> Signup and view all the answers

    Study Notes

    Finite Square Well Potential

    • The finite square well potential has two physically interesting cases: E > V0 and 0 < E < V0.
    • The solutions to the Schrödinger equation for these cases yield a continuous doubly-degenerate energy spectrum for E > V0 and a discrete nondegenerate spectrum for 0 < E < V0.

    Scattering Solutions (E > V0)

    • Classically, a particle with momentum 2m(E - V0) will speed up to 2mE between -a/2 and a/2, and then slow down to its initial momentum in the region x > a.
    • All particles coming from the left will be transmitted, and none will be reflected back; therefore, T = 1 and R = 0.
    • The solutions to the Schrödinger equation in the regions x < -a/2 and x > a/2 are O1(x) = Aeik1x and O3(x) = De-k1x, respectively.
    • The solutions are either antisymmetric (odd) or symmetric (even) under space inversion.

    Bound States (0 < E < V0)

    • The eigenvalues are determined by the continuity conditions at x = ±a/2.
    • There exist three bound states: the ground state (even state), the first excited state (odd state), and the second excited state (even state).
    • The well width at which n states are allowed is given by R ≈ (n + 1)²π²/(2ma²).

    Graphical Solutions

    • The graphical solutions for the finite square well potential are given by the intersections of R² - :n² with :n tan :n and -:n cot :n.
    • The graphical solutions are used to determine the number of bound states and the corresponding energies.

    Example 4.2

    • The number of bound states and the corresponding energies for the finite square well potential can be determined by solving the graphical equation.
    • For R = 1, there is only one bound state with energy E0 ≈ 1.1ℏ²/(ma²).

    Harmonic Oscillator

    • The Hamiltonian of a particle oscillating with an angular frequency ω under a one-dimensional harmonic potential is H = (P²/2m) + (mω²X²/2).
    • The energy eigenvalues and eigenstates of this Hamiltonian can be studied using two methods: the analytic method (solving the time-independent Schrödinger equation) and the ladder or algebraic method (using operator algebra involving creation and annihilation operators).

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    Description

    Explore the concept of the finite square well potential in quantum mechanics, including its graphical representation and mathematical equations. Understand the role of V0 and E in the potential energy diagram.

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