IB DP Physics: HL复习笔记12.1.7 Quantization of Angular Momentum

Quantization of Angular Momentum

  • Angular momentum is a property of any spinning or rotating body, very similar to linear momentum
    • In linear motion, momentum is the product of mass and velocity
  • In rotational motion the momentum is the product of moment of inertia and angular speed
  • Angular momentum is a vector, this means:
    • The magnitude is equal to the momentum of the particle times its radial distance from the centre of its circular orbit
    • The direction of the angular momentum vector is normal to the plane of its orbit with the direction being given by the corkscrew rule

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Angular momentum acts at right angles to the direction of rotation

  • For an electron moving in a straight line, the matter wave takes a familiar wave shape consisting of peaks and troughs
    • Although the electron itself isn't oscillating up and down, only the matter wave

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de Broglie matter wave for an electron moving in a straight line at constant speed

  • For the same electron moving in a circle, the matter wave still has a sinusoidal shape but is wrapped into a circle

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de Broglie matter wave for an electron moving in a circular orbit at constant speed

  • As the electron continues to orbit in a circle two possibilities may occur:

1. On completing one oscillation, the waves overlap in phase

    • The waves will continue in phase over many orbits giving rise to constructive interference and a standing wave

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de Broglie matter wave where 3λ is less than the orbits circumference

2. On completing one oscillation, the waves overlap but they are not in phase
    • In other words, peak overlaps with peak, trough with trough
    • This means that where the waves overlap, destructive interference occurs and as a result, no such electron orbit is allowed

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de Broglie matter wave where n = 3. Here the circumference of the circular orbit is 3λ

Bohr Condition

  • The Bohr Condition is given by the relation:

Worked Example

Determine the velocity of the electron in the first Bohr orbit of the hydrogen atom (n = 1).

You may use the following values:

  • Mass of an electron = 9.1 × 10−31 kg
  • Radius of the orbit = 0.529 × 10−10 m
  • Planck's constant = 6.63 × 10−34 kg m2 s-1

Step 1: List the known quantities

    • Mass of an electron, m = 9.1 × 10−31 kg
    • Radius of the orbit, r = 0.529 × 10−10 m
    • Planck's constant, h = 6.63 × 10−34 kg m2 s-1

Step 2: Write the Bohr Condition equation and rearrange for velocity, v

 

 

 

 

 

 

 

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