Questions on WKB approximation

[jasum]

The description in P.252in liboff's quantum mechanics,

I cannot not figure out the continuity and continue in first order derivative of the wave function
$$\varphi_I = \frac{1}{\sqrt{\kappa}} \exp {(\int_{x_1}^{x} \kappa dx)}$$

in (7.184)
$$\varphi_{\amalg} = \frac{2}{\sqrt{k}} \exp {(\int_{x_1}^{x} k dx + \pi/4)}$$

and (7.185)

The potenial is
$$V(x) = E - F_1 (x-x_1)$$

F1 is a constant
where the boundary is in x1
Thank you!

 

[Jhero]

Thank you for your question regarding the continuity and first order derivative of the wave function in P.252 of Liboff's Quantum Mechanics.

Firstly, it is important to understand that the wave function, denoted by the symbol ψ, describes the behavior of a quantum system. In quantum mechanics, the wave function is a complex-valued function that contains all the information about the system, including its position, momentum, and energy. The continuity and first order derivative of the wave function are crucial in understanding the behavior of the system and how it evolves over time.

In equation (7.184), the wave function ψ_I is given as a function of the spatial variable x, with a constant κ in the denominator. This form of the wave function is known as the "incident wave" and describes the behavior of a wave approaching a potential barrier. The term in the exponential, ∫x1x κ dx, represents the phase of the wave and is related to the momentum of the particle. The continuity of this wave function at x=x1 ensures that the wave is continuous as it approaches the potential barrier.

In equation (7.185), the wave function ψ_∏ is given as a function of the spatial variable x, with a constant k in the denominator. This form of the wave function is known as the "transmitted wave" and describes the behavior of a wave that has passed through a potential barrier. The term in the exponential, ∫x1x k dx + ∏/4, represents the phase of the wave and is related to the momentum of the particle. The continuity of this wave function at x=x1 ensures that the wave is continuous as it passes through the potential barrier.

The potential V(x) = E - F_1 (x-x_1) in both equations represents the energy of the particle, with E being the total energy and F_1 being a constant. The boundary at x=x1 in both equations indicates that the potential barrier starts at this point.

In summary, the continuity and first order derivative of the wave function are essential in understanding the behavior of a quantum system, particularly when it interacts with a potential barrier. I hope this helps clarify your understanding. Please let me know if you have any further questions.