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7:08 PM M \( 11^{46} .11 .\left..1\right|^{31.3} \) 87 Edit Easily add text in PDF Add It so happens that every solution to the (time-dependent) Schrödinger equation can be written in this form-it is simply a matter of finding the right constants \( \left(c_{1}, c_{2}, \ldots\right) \) so as to fit the initial conditions for the problem at hand. You'll see in the following sections how all this works out in practice, and in Chapter 3 we'll put it into more elegant language, but the main point is this: Once you've solved the time-independent \( { }^{5} \triangle \) linear combination of the functions \( f_{1}(z), f_{2}(z), \ldots \) is an expression of the form \[ f(z)=c_{1} f_{1}(z)+c_{2} f_{2}(z)+\cdots \] where \( c_{1}, c_{2}, \ldots \) are any (complex) constants. hap. 2 The Time-Independent Schrödinger Equation Schrödinger equation, you're essentially done; getting from there to the general solution of the time-dependent Schrödinger equation is simple and straightforward. Problem 2.1 Prove the following theorems: (a) For normalizable solutions, the separation constant \( E \) must be real. Hint: Write \( E \) (in Equation 2.6) as \( E_{0}+i \Gamma \) (with \( E_{0} \) and \( \Gamma \) real), and show that if Equation 1.20 is to hold for all \( t, \Gamma \) must be zero. (b) \( \psi \) can always be taken to be real (unlike \( \Psi \), which is necessarily complex). Note: This doesn't mean that every solution to the timc-independent Schrödinger equation is real; what it says is that if you've got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So in Equation 2.14 you might as well stick to \( \psi \) 's that are real. Hint: If \( \psi(x) \) satisfies the time-independent Schrödinger equation for a given \( E \), so too does its complex conjugate, and hence also the real linear combinations \( \left(\psi+\psi^{}\right) \) and \( i\left(\psi-\psi^{}\right) \). (c) If \( V(x) \) is an even function [i.e., \( V(-x)=V(x) \) ], then \( \psi(x) \) can always be taken to be either even or odd. IIint: If \( \psi(x) \) satisfics the time-independent Schrödinger equation for a given \( E \), so too does \( \psi(-x) \), and hence also the even and odd linear combinations \( \psi(x) \pm \psi(-x) \). \( \overline{\text { Problem 2.2 Show that } E \text { must exceed the minimum value of } V(x) \text { for every }} \) normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.4 in the form \[ \frac{d^{2} \psi}{d x^{2}}=\frac{2 m}{\hbar^{2}}[V(x)-E] \psi \] if \( E<V_{\min } \), then \( \psi \) and its second derivative always have the same sign-argue that such a function cannot be normalized. THE INFINITE SOUARE WELL

          7:08 PM M
\( 11^{46} .11 .\left..1\right|^{31.3} \)
87
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Easily add text in PDF
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It so happens that every solution to the (time-dependent) Schrödinger equation can be written in this form-it is simply a matter of finding the right constants \( \left(c_{1}, c_{2}, \ldots\right) \) so as to fit the initial conditions for the problem at hand. You'll see in the following sections how all this works out in practice, and in Chapter 3 we'll put it into more elegant language, but the main point is this: Once you've solved the time-independent
\( { }^{5} \triangle \) linear combination of the functions \( f_{1}(z), f_{2}(z), \ldots \) is an expression of the form
\[
f(z)=c_{1} f_{1}(z)+c_{2} f_{2}(z)+\cdots
\]
where \( c_{1}, c_{2}, \ldots \) are any (complex) constants.
hap. 2 The Time-Independent Schrödinger Equation
Schrödinger equation, you're essentially done; getting from there to the general solution of the time-dependent Schrödinger equation is simple and straightforward.
*Problem 2.1 Prove the following theorems:
(a) For normalizable solutions, the separation constant \( E \) must be real. Hint: Write \( E \) (in Equation 2.6) as \( E_{0}+i \Gamma \) (with \( E_{0} \) and \( \Gamma \) real), and show that if Equation 1.20 is to hold for all \( t, \Gamma \) must be zero.
(b) \( \psi \) can always be taken to be real (unlike \( \Psi \), which is necessarily complex). Note: This doesn't mean that every solution to the timc-independent Schrödinger equation is real; what it says is that if you've got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So in Equation 2.14 you might as well stick to \( \psi \) 's that are real. Hint: If \( \psi(x) \) satisfies the time-independent Schrödinger equation for a given \( E \), so too does its complex conjugate, and hence also the real linear combinations \( \left(\psi+\psi^{*}\right) \) and \( i\left(\psi-\psi^{*}\right) \).
(c) If \( V(x) \) is an even function [i.e., \( V(-x)=V(x) \) ], then \( \psi(x) \) can always be taken to be either even or odd. IIint: If \( \psi(x) \) satisfics the time-independent Schrödinger equation for a given \( E \), so too does \( \psi(-x) \), and hence also the even and odd linear combinations \( \psi(x) \pm \psi(-x) \).
\( * \overline{\text { Problem 2.2 Show that } E \text { must exceed the minimum value of } V(x) \text { for every }} \) normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.4 in the form
\[
\frac{d^{2} \psi}{d x^{2}}=\frac{2 m}{\hbar^{2}}[V(x)-E] \psi
\]
if \( E<V_{\min } \), then \( \psi \) and its second derivative always have the same sign-argue that such a function cannot be normalized.
THE INFINITE SOUARE WELL

7:08 PM M
11^46 .11 ...1|^31.3
87
Edit
Easily add text in PDF
Add
It so happens that every solution to the (time-dependent) Schrödinger equation can be written in this form-it is simply a matter of finding the right constants (c1, c2, …) so as to fit the initial conditions for the problem at hand. You'll see in the following sections how all this works out in practice, and in Chapter 3 we'll put it into more elegant language, but the main point is this: Once you've solved the time-independent
^5 linear combination of the functions f1(z), f2(z), … is an expression of the form

f(z)=c1 f1(z)+c2 f2(z)+⋯

where c1, c2, … are any (complex) constants.
hap. 2 The Time-Independent Schrödinger Equation
Schrödinger equation, you're essentially done; getting from there to the general solution of the time-dependent Schrödinger equation is simple and straightforward.
*Problem 2.1 Prove the following theorems:
(a) For normalizable solutions, the separation constant E must be real. Hint: Write E (in Equation 2.6) as E0+i Γ (with E0 and Γ real), and show that if Equation 1.20 is to hold for all t, Γ must be zero.
(b) ψ can always be taken to be real (unlike Ψ, which is necessarily complex). Note: This doesn't mean that every solution to the timc-independent Schrödinger equation is real; what it says is that if you've got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So in Equation 2.14 you might as well stick to ψ 's that are real. Hint: If ψ(x) satisfies the time-independent Schrödinger equation for a given E, so too does its complex conjugate, and hence also the real linear combinations (ψ+ψ^*) and i(ψ-ψ^*).
(c) If V(x) is an even function [i.e., V(-x)=V(x) ], then ψ(x) can always be taken to be either even or odd. IIint: If ψ(x) satisfics the time-independent Schrödinger equation for a given E, so too does ψ(-x), and hence also the even and odd linear combinations ψ(x) ±ψ(-x).
* Problem 2.2 Show that E must exceed the minimum value of V(x) for every normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.4 in the form

(d^2ψ)/(d x^2)=(2 m)/(ħ^2)[V(x)-E] ψ

if E<Vmin, then ψ and its second derivative always have the same sign-argue that such a function cannot be normalized.
THE INFINITE SOUARE WELL

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