Chapter 3: Q13P (page 112)

(a) Prove the following commutator identity:
$[A B . C] = A [B . C] + [A . C] B$
b) Show that
$[x^{n}, p] = i h n x^{n - 1}$
(c) Show more generally that
$[f (x), p] = i h \frac{d f}{d x}$
for any function $f (x)$ .

Short Answer

Expert verified

a) $[AB . C] = A [B . C] + [A . C] B$

b) $[x^{n}, p] = i h n x^{n - 1}$

c) $[f (x), p] = i h \frac{df}{dx}$

Step by step solution

Concept used

Commutator of two quantities and is defined:

$[A, B] = A B - B A$

$\Rightarrow [A B, C] = A B C - C A B$ ……. (1)

Prove Commutator quantity

We can add $(A C B - A C B)$ to equation (1):

localid="1658124315043" $\begin{array}{rcl} [A B, C] & = & A B C - C A B + A C B - A C B \\ = & A (B C - C B) + (A C - C A) B \\ = & A [B, C] + [A, C] B \\ = & A [B, C] + [A, C] B \end{array}$

Use mathematical induction

We use mathematical induction:

Mathematical induction:

$n = 1 [x, p] = i h$

We assume that following identity is valid for some $n$ :

$[x^{n}, p] = i h n x^{n - 1}$

Induction step: $n + 1$

localid="1658124506145" $\begin{array}{rcl} [x^{n + 1}, p] & = & [x \cdot x^{n} p] \\ = & [x^{n}, p] + [x, p] x^{n} \\ = & x \cdot i h n x^{n - 1} + i h x^{n} \\ = & i h n x^{n} + i h x^{n} \\ = & i h (n + 1) x^{n} \end{array}$

To prove the given relation, after commutator have a test function

After commutator have a test function:

$\begin{array}{rcl} [f (x), p] ψ (x) & = & [f (x), - i h \frac{d}{d x}] ψ (x) \\ = & - i h [f (x), \frac{d}{d x}] ψ (x) \\ = & - i h (f \frac{d ψ}{d x} - \frac{d}{d x} (f (x) ψ (x))) \\ = & - i h (f \frac{d ψ}{d x} - f \frac{d ψ}{d x} - \frac{d f}{d x} ψ) \\ = & i h \frac{d f}{d x} ψ . \end{array}$

Since this relation must be valid for any test function $ψ (x)$ , it follows:

$[f (x), p] = i h \frac{d f}{d x}$

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(a) Prove the following commutator identity:
$[A B . C] = A [B . C] + [A . C] B$
b) Show that
$[x^{n}, p] = i h n x^{n - 1}$
(c) Show more generally that
$[f (x), p] = i h \frac{d f}{d x}$
for any function $f (x)$ .

Short Answer

Step by step solution

Concept used

Prove Commutator quantity

Use mathematical induction

To prove the given relation, after commutator have a test function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Study anywhere. Anytime. Across all devices.

(a) Prove the following commutator identity:[AB.C]=A[B.C]+[A.C]Bb) Show that[xn,p]=ihnxn-1(c) Show more generally that[f(x),p]=ihdfdxfor any functionf(x).

Short Answer

Step by step solution

Concept used

Prove Commutator quantity

Use mathematical induction

To prove the given relation, after commutator have a test function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Electricity

Physical Quantities And Units

Electricity and Magnetism

Torque and Rotational Motion

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

(a) Prove the following commutator identity:
$[A B . C] = A [B . C] + [A . C] B$
b) Show that
$[x^{n}, p] = i h n x^{n - 1}$
(c) Show more generally that
$[f (x), p] = i h \frac{d f}{d x}$
for any function $f (x)$ .