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?

Even (B) Signup and view all the answers

How many bound states are there in the given system?

Three (D) Signup and view all the answers

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

2 (D) Signup and view all the answers

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

Either even or odd (D) Signup and view all the answers

What is the condition required to determine the eigenvalues?

The continuity conditions at x = a/2 (A) Signup and view all the answers

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

They are either symmetric or antisymmetric (D) Signup and view all the answers

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

The intersections of R^2 :n2 with :n tan :n and :n cot :n (D) Signup and view all the answers

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

They grow exponentially (B) Signup and view all the answers

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

x < a/2 (A) Signup and view all the answers

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

R = 1 (C) Signup and view all the answers

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

Csin(k2x) - De^(k1x) (B) Signup and view all the answers

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

n = 0 (C) Signup and view all the answers

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

:0 tan :0 = 1 + :02 (C) Signup and view all the answers

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

x > a/2 (B) Signup and view all the answers

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

:0 = 0.73909 (B) Signup and view all the answers

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

Bcos(k2x) - De^(k1x) (A) Signup and view all the answers

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

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

What is the condition for having only one bound state?

R = 1 (D) Signup and view all the answers

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

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

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

V(x) = 1/2 mω²x² (C) Signup and view all the answers

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

The equation is difficult to solve due to the complexity of the potential (B) Signup and view all the answers

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

To find the energy eigenvalues and eigenstates of the Hamiltonian (A) Signup and view all the answers

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

It is more straightforward, more elegant, and simpler (D) Signup and view all the answers

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

They are used to express the various quantities in terms of matrices (A) Signup and view all the answers

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

It is more straightforward and simpler (C) Signup and view all the answers

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

E > V0 (C) Signup and view all the answers

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

It increases (A) Signup and view all the answers

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

0 (D) Signup and view all the answers

What is the direction of the incident wave?

From left to right (C) Signup and view all the answers

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

x > a/2 (A) Signup and view all the answers

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

E < V0 (B) Signup and view all the answers

What is the momentum of the particle initially?

$ ext{\sqrt}{2mE}$ (D) Signup and view all the answers

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

1 (D) Signup and view all the answers

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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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