Prove that the solution of Schrodinger equation can be expressed in the form ψ ( x , t ) = ∫ K ( x , x ′ , t ) ψ ( x ′ , 0 ) d x ′ .

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Answer 1

Question

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

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Consider the Schrodinger equation for a one-dimensional linear harmonic oscillator: -(hbar2/2m) * d2ψ/dx2 + (kx2/2)*ψ(x) = Eψ(x) Substitute the wavefunction ψ(x) = e-(x^2)/(ξ^2) and find ξ and E required to satisfy the Schrodinger equation. [Hint: First calculate the second derivative of ψ(x), then substitute ψ(x) and ψ′′(x). After this substitution, there will be an overall factor of e-(x^2)/(ξ^2) on both sides of the equation which canbe an canceled out. Then, gather all terms which depend on x into one coefficient multiplying x2. This coefficient must be zero because the equation must be satisfied for any x, and equating it with zero yields the expression for ξ. Finally, the remaining x-independent part of the equation determines the eigenvalue for energy E associated with this solution.]

a) Write down the one-dimensional time-dependent Schro ̈dinger equation for a particle of mass m described by a wavefunction Ψ(t, x) in a potential V (x). b) For energy eigenstates, the wavefunction can be written as Ψ(t, x) = f (t) ψ(x). For this wavefunction:(i) state the time-independent equation that must be satisfied by ψ(x). (ii) derive an expression for f(t), in terms of the energy of the particle E.

A function of the form e^−gx2 is a solution of the Schrodinger equation for the harmonic oscillator, provided that g is chosen correctly. In this problem you will find the correct form of g. (a) Start by substituting Ψ = e^−gx2 into the left-hand side of the Schrodinger equation for the harmonic oscillator and evaluating the second derivative. (b) You will find that in general the resulting expression is not of the form constant × Ψ, implying that Ψ is not a solution to the equation. However, by choosing the value of g such that the terms in x^2 cancel one another, a solution is obtained. Find the required form of g and hence the corresponding energy. (c) Confirm that the function so obtained is indeed the ground state of the harmonic oscillator and has the correct energy.

Answer 2

Question

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

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Answer 3

Question

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

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Answer 4

Question

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

Expert Solution & Answer

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Answer 5

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

Answer 6

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

Answer 7

Chapter 6.8, Problem 6.30P

(a)

To determine

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

Answer 8

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

Answer 9

(b)

To determine

Obtain the function K(x,x′,t) for the particle.

Answer 10

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

Answer 11

(c)

To determine

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

Answer 12

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

Answer 13

(d)

To determine

Obtain the function K(x,x′,t) for the particle when the states are continuous.

Answer 14

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

Answer 15

(e)

To determine

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

Answer 16

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Answer 17

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Answer 18

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Answer 19

Ch. 6.1 - Prob. 6.1P Ch. 6.2 - Prob. 6.2P Ch. 6.2 - Prob. 6.3P Ch. 6.2 - Prob. 6.4P Ch. 6.2 - Prob. 6.5P Ch. 6.2 - Prob. 6.7P Ch. 6.4 - Prob. 6.8P Ch. 6.4 - Prob. 6.9P Ch. 6.4 - Prob. 6.10P Ch. 6.4 - Prob. 6.11P

Prove that the solution of Schrodinger equation can be expressed in the form ψ ( x , t ) = ∫ K ( x , x ′ , t ) ψ ( x ′ , 0 ) d x ′ .

Solution Summary: The author proves that the solution of Schrodinger equation can be expressed in the form psi(x,t)=displaystyleint

Prove that the solution of Schrodinger equation can be expressed in the form ψ(x,t)=∫K(x,x′,t)ψ(x′,0)dx′.

Obtain the function K(x,x′,t) for the particle.

Obtain ψ(x,t) for the particle in part (a) for the given state and compare with the solution of problem 2.49.

Obtain the function K(x,x′,t) for the particle when the states are continuous.

Obtain the function ψ(x,t) for the free particle with initial state ψ(x,0)=(2aπ)14e−ax2 and compare the result with problem 2.21.

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