QM solutions to the Schrodinger Equation

In summary, the conversation discusses the solutions to a potential function with different forms for the wavefunction in different sections. The solutions involve either trigonometric functions or complex exponentials, with the latter being more practical due to ease of manipulation. The identities for complex exponentials are also mentioned.
  • #1
Maybe_Memorie
353
0

Homework Statement



Here's something that's confusing me. Say we have a potential

V(x) = Vo if x < 0, x > a
and
V(x) = 0 if 0 < x < a

(yes I know the notation with greater than/equals etc isn't totally correct, but you know what I'm talking about.)

In the middle section, ψ'' + k2ψ = 0

Whenever I see the solutions it's always ψ = Asin(kx) + Bcos(kx) in the middle section where k2 = -2mE/h2

In the right/left sections, ψ'' + f2ψ = 0, with f2 = -2m(E - V0)/h2

The solutions here always seem to be complex exponentials.

Can someone please explain the difference to me in the solutions?
 
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  • #2
You must know the identities:
exp(±iø) = cosø ± i*sinø
So the two sets of solutions you're referring to are basically equivalent once you take the real part, only the exponential form turns out to be more practical in most cases because easier to manipulate.
 

FAQ: QM solutions to the Schrodinger Equation

1. What is the Schrodinger Equation?

The Schrodinger Equation is a fundamental equation in quantum mechanics that describes how the state of a quantum system changes over time. It is a differential equation that relates the time evolution of a quantum state to the energy of the system.

2. What are QM solutions to the Schrodinger Equation?

QM solutions to the Schrodinger Equation refer to the set of mathematical solutions that satisfy the equation and describe the behavior of a quantum system. These solutions are typically in the form of wavefunctions that represent the probability amplitude of finding a particle in a certain state at a given time.

3. How are QM solutions to the Schrodinger Equation used in real-world applications?

QM solutions to the Schrodinger Equation are used in a wide range of real-world applications, including quantum computing, quantum chemistry, and materials science. They help us understand and predict the behavior of particles at the atomic and subatomic level, and have led to advancements in technology and materials.

4. What are the limitations of QM solutions to the Schrodinger Equation?

One limitation of QM solutions to the Schrodinger Equation is that they cannot fully describe the behavior of particles at high energies or in extreme conditions, such as in black holes. Additionally, the solutions may not accurately predict the behavior of macroscopic objects, as they are based on probabilistic principles rather than deterministic ones.

5. Are there alternative theories to QM solutions to the Schrodinger Equation?

Yes, there are alternative theories to QM solutions to the Schrodinger Equation, such as the pilot wave theory and the many-worlds interpretation. These theories propose different explanations for the behavior of quantum systems and are still under debate in the scientific community.

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