Quantum Mechanics - Time dependent solution - x's and t's not mixed up

In summary, the topic discusses the time-dependent solutions of quantum mechanics, emphasizing the importance of clearly distinguishing between spatial variables (x's) and time variables (t's) in the equations. It highlights the implications of this distinction for understanding quantum states and their evolution over time, ultimately reinforcing the foundational principles of quantum theory.
  • #1
laser
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Homework Statement
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Relevant Equations
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Screenshot_6.png


Next, we assume a solution in this form:
Screenshot_7.png


Which simplifies (according to my notes) to this:
Screenshot_8.png


In the middle equation, we have factorised out the F(t). My question is why is it wavefunction(x,t) rather than wavefunction(x). I first thought it was a mistake in the notes, but it uses the same equation later on.

Edit: And F(t) on the rightmost equation.
 
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  • #2
Probably just a typo. It should be ##\psi(x)##. Also, ##F(t)## is missing in the last term on the right in the last equation.
 
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  • #3
DrClaude said:
Probably just a typo. It should be ##\psi(x)##. Also, ##F(t)## is missing in the last term on the right in the last equation.
Cheers - yeah I forgot to mention that here but you're right
 

FAQ: Quantum Mechanics - Time dependent solution - x's and t's not mixed up

What is the time-dependent Schrödinger equation?

The time-dependent Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is represented as: \[ i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \hat{H} \Psi(x, t) \]where \( \Psi(x, t) \) is the wave function of the system, \( \hat{H} \) is the Hamiltonian operator, and \( \hbar \) is the reduced Planck's constant.

What does the wave function represent in quantum mechanics?

The wave function, denoted as \( \Psi(x, t) \), contains all the information about a quantum system. The absolute square of the wave function, \( |\Psi(x, t)|^2 \), gives the probability density of finding a particle at position \( x \) at time \( t \). It encapsulates both the position and the momentum properties of the particle.

How do we separate variables in the time-dependent Schrödinger equation?

To separate variables in the time-dependent Schrödinger equation, we assume a solution of the form \( \Psi(x, t) = \psi(x) \phi(t) \). Substituting this into the equation allows us to separate it into two independent equations: one that depends only on position \( x \) and another that depends only on time \( t \). This leads to the time-independent Schrödinger equation for \( \psi(x) \) and a simple differential equation for \( \phi(t) \).

What is the significance of the time-independent Schrödinger equation?

The time-independent Schrödinger equation is used to find the stationary states of a quantum system, which are states where the probability distribution does not change over time. It is given by:\[ \hat{H} \psi(x) = E \psi(x) \]where \( E \) is the energy eigenvalue associated with the state \( \psi(x) \). Solving this equation provides the allowed energy levels and corresponding wave functions for a quantum system.

How do boundary conditions affect the solutions in quantum mechanics?

Boundary conditions are crucial in determining the allowed solutions to the Schrödinger equation. They specify the behavior of the wave function at the edges of the system, such as potential barriers or confinement in a box. These conditions can lead to quantization of energy levels, as only certain wave functions that satisfy the boundary conditions are permissible, thus influencing the physical properties of the quantum system.

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