36 Questions
What is the well width at which n states are allowed in a bound state?
RH = √(2n²h²/V₀) / (2ma²)
What is the sequence of states in the spectrum of a bound state?
Even, odd, even, odd, ...
What is the parity of the first excited state?
Odd
What is the parity of the ground state?
Even
How many bound states are there in the given system?
Three
What is the value of n for the second excited state?
2
What type of eigenfunctions do symmetric one-dimensional Hamiltonians have?
Either even or odd
What is the condition required to determine the eigenvalues?
The continuity conditions at x = a/2
What can be said about the solutions of (4.90) to (4.92)?
They are either symmetric or antisymmetric
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
What happens to the physically unacceptable solutions for large values of x?
They grow exponentially
What is the condition for x in equation (4.93)?
x < a/2
What is the value of R when there is only one bound state?
R = 1
What is the expression for Oa(x) in equation (4.95)?
Csin(k2x) - De^(k1x)
What is the value of n corresponding to the bound state when R = 1?
n = 0
What is the equation that determines the energy of the bound state?
:0 tan :0 = 1 + :02
What is the condition for x in equation (4.94)?
x > a/2
What is the numerical solution of cos2 :0 = :02?
:0 = 0.73909
What is the expression for Os(x) in equation (4.96)?
Bcos(k2x) - De^(k1x)
What is the expression for the energy of the bound state?
ma^2 E0 = 0.73909 h^2 / (2m)
What is the condition for having only one bound state?
R = 1
What is the expression for R^2 in terms of m, V0, and h?
R^2 = m^2 V0 / (h^2)
What is the form of the potential in the Hamiltonian described?
V(x) = 1/2 mω²x²
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
What is the purpose of the ladder or algebraic method in this context?
To find the energy eigenvalues and eigenstates of the Hamiltonian
Why is the ladder or algebraic method preferred over the analytic method?
It is more straightforward, more elegant, and simpler
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
What is the advantage of the matrix formulation in the ladder or algebraic method?
It is more straightforward and simpler
What is the condition for the energy spectrum to be continuous and doubly-degenerate?
E > V0
What happens to the particle's momentum between -a/2 and a/2?
It increases
What is the value of R (reflection coefficient) in the region x < -a/2?
0
What is the direction of the incident wave?
From left to right
What is the range of x where the particle slows down?
x > a/2
What is the condition for the energy spectrum to be discrete and non-degenerate?
E < V0
What is the momentum of the particle initially?
$ ext{\sqrt}{2mE}$
What is the value of T (transmission coefficient) in the region x < -a/2?
1
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).
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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