Solving Schrodinger's Equation Homework

  • Thread starter oddiseas
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In summary, the given eigenfunction is described as \Psi=iM\exp(-x/2) for x greater than or equal to zero and 0 for x less than zero. The corresponding wave function can be expressed as \Psi(x,t)=\sqrt{2}ie^{-x}(ik+1/2)g(t) and the eigenfunctions can be expressed as f_n(x)=\sqrt{2}ie^{-x}(ik+1/2). The normalizing constants are M=1 and A=\sqrt{2}. The wavefunction approaches zero as x approaches infinity.
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oddiseas
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Homework Statement



At t=0 a particle is described by the eigenfunction:

[tex]\Psi[/tex]= i[tex] M [/tex] [tex]e^{\frac{-x}{2}}[/tex] x [tex]\geq 0[/tex]
0 if x [tex]\prec 0[/tex]

a) Write an expression for the corresponding wave function

b) find the epression for the eigenfunctions.



Homework Equations





The Attempt at a Solution



Does the wavefunction always approach zero as x approaches infinity?

if so this gives me:
f(x)=Be^ikx+Ce^-ikx
f(0)=Aie^(-x/2)
f([tex]\infty[/tex])=0 then B=0
f(x)=Aie^(-x/2)e^-ikx

f(x)=Aie^-x(ik+1/2)

then normalising this solution gives A=[tex]\sqrt{2}[/tex]

f[tex]_{n}[/tex](x)=[tex]\sqrt{2}[/tex]ie^-x(ik+1/2)

then normalising the initial condition give M=1.

[tex]\Psi[/tex]= [tex]\sum[/tex] A*[tex]\sqrt{2}[/tex]ie^-x(ik+1/2)*g(t)

This is as far as i could get;
 
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  • #2
oddiseas said:

Homework Statement



At t=0 a particle is described by the eigenfunction:

[tex]\Psi= i M \exp\left({\frac{-x}{2}}\right)[/tex] [tex]x\geq 0[/tex]
0 if [tex]x< 0[/tex]

The Attempt at a Solution



Does the wavefunction always approach zero as x approaches infinity?

With this eigenfunction, yes. [itex]\lim_{x\rightarrow\infty}\exp(-x)=0[/itex]
 

Related to Solving Schrodinger's Equation Homework

1. What is Schrodinger's Equation?

Schrodinger's equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It is a partial differential equation that determines the wave function of a quantum system, which contains all the information about the system's physical properties.

2. What is the purpose of solving Schrodinger's Equation?

The purpose of solving Schrodinger's equation is to understand the behavior and properties of quantum systems. By solving the equation, scientists can predict the probability of finding a particle at a certain location and time, as well as other physical properties of the system.

3. How do you solve Schrodinger's Equation?

Schrodinger's equation can be solved using various mathematical methods, such as separation of variables, perturbation theory, and numerical methods. The solution is typically in the form of a wave function, which can then be used to calculate the desired physical properties of the quantum system.

4. What are some real-world applications of Schrodinger's Equation?

Schrodinger's equation has many applications in various fields, including quantum chemistry, materials science, and condensed matter physics. It is used to understand and predict the behavior of atoms, molecules, and materials at the quantum level, which is essential for developing new technologies and materials.

5. What are the challenges in solving Schrodinger's Equation?

Solving Schrodinger's equation can be challenging due to its complex mathematical nature and the limitations of existing numerical methods. The equation also becomes more complicated for systems with more than one particle, making it difficult to find exact solutions. In addition, the interpretation of the wave function and the concept of quantum superposition can also be challenging for some individuals.

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